A lot of replies has been posted since I started compiling this
TL;DR version at the end.
Unfortunately not. With this additional rule we now a situation where we need to determine what does casting a spell within antimagic field means. I postulate that casting a spell is an action (usually standard) and not a Special Ability. Because Antimagic field description does not say that actions (to cast a spell) are suppressed then they are not. Actions and Special Abilities are quite distinct of each other and not to be confused with each other.
Therefore antimagic cannot be used as a proof that casting a spell is an extraordinary ability.
Check here again, specifically the table at the top: http://www.d20srd.org/srd/specialAbilities.htm
Note that spell like abilities are suppressed (not just the actions... the abilities). However, the ability to cast spells is not suppressed (just the action). Likewise, from the link given: "Supernatural abilities are magical and go away in an antimagic field." So again, if spells were a supernatural or spell like ability, you couldn't use them at all. But the abilities aren't suppressed (as they would be if they were Su or Sp), just the actions... so the abilities must be Na or Ex (and since they're Special Abilities, they can't be Na either, and must be Ex).
Also, don't forget this quote, from the Antimagic spell itself: "The space within this barrier is impervious to most magical effects, including spells, spell-like abilities, and supernatural abilities." Spells are clearly separate from Spell Like Abilities and Supernatural Abilities.
While we're at it, check out Fiendish Codex II (I've been looking through that one a lot, since it came up). Many feats in there share similar language with antimagic... they constantly refer to "spell, spell like ability, or supernatural ability." Clearly, these three things are separate.
The SRD sadly is not a valid source for determining what Spellcasting is nor what
Special Attack: Spells is. I maintain that ability to make an action to cast a spell is not a
Special Ability. Hence it has no type.
Who can abjudicate the conflict that Core Rulebook PHB, which states explicitly that Spells are not extraordinary abilities, and other sources MM5 Special Quality: Arcane Talent (Ex), Core Rulebook MM1 Special Attack: Spells, etc are not fully agreeing with each other? Who can say which part have more weigth than the other? I would call it RAI if one type is chosen instead of the other, when no type is explicitly stated.
Primary Source rules come into play here. MM1 is the primary source for Special Abilities.
PHB, DMG and MM1 + erratas are imho the best sources for this issue. For
Class Feature: Spells the PHB is the primary source. For
Special Attack: Spells the PHB is the primary source (for a small portion the MM1 is the primary source).
...
But we can't really do that. Supernatural and extraordinary are out explicitly for the same reasons. Spell-like clearly makes a distiction between spell-like and spells. So it can't be spell-like either. Spells is not a physical in nature, so it can't be natural either. Ergo, we have ruled out all of the four Special ability types.
Fortunately if you closely read the Specal Ability quote you'll notice that Special Abilites are meant for magical effects without being a spellcaster. So we have way out. Spellcaster's (like wizard) Class Feature: Spells is something different than a Special Ability.
Right logic, wrong conclusion. Magical class features which are not spellcasting are Sp or Su... but this whole quote is talking about non spellcasting, so it doesn't apply to spellcasters at all.
Also, Class Feature spellcasting is listed as a Special Ability in multiple places, including the SRD (under Special Abilities) and the Monster Manual (Celestial Chargers, which are under unicorns if you'd like to check and are just Unicorns that are Clerics). In fact, you'll find that all creatures that get class based spellcasting have their spells listed as Special Attacks, which makes them indeed Special Abilities.
I suppose you mean
Class Feature: Spells instead of
Class Feature: Spellcasting as there is no such thing. SRD cannot be use as an evidence as it is not a valid source. I checked the Celestial Charger and yes it says that it has Cleric levels and that it has
Special Attack: Spells. Is
Special Attack: Spells a
Special Ability? No it is not. I see the contrary.
Few rules quotes ahead:
First sentence says
unusual abilities. Now, all those abilities are not necessarily
Special Abilities as it is not explicitly stated so.
Then these
(unusual) abilities are divided between
Special Attacks and
Special Qualities.
Then
Special Qualities is broken down to defenses, vulnerabilities and other special abilities (not modes of attacks). So Special Quality: [Defense] and Special Quality: [Vulnerability] are not necessarily
Special Abilities.
But
Special Attacks is not broken down. It is only implied via
Special Qualities that special abilities (that are modes of attack) belong here. So there can be
(unusual) abilities that are
Special Attacks but are not
Special Abilities.
Special Attack: Spells fits to this category nicely.
Then we have the glossary in the MM1. It separates
Special Abilities and
Spells from each other as they have their own subchapters. That alone implies that
Spells is something different than
Special Abilities.
Special Attack: Spells cannot be extraordinary because they are magical, become suppressed or fail in antimagic field, and can be disrupted (e.g. casting a spell in combat).
Nor it can be spell-like or supernatural. I'll leave out rules quotes as I assume this is not disputed. In fact
Special Attack: Spells is not a
Special Ability.
Here it says that "can cast" i.e. the generic ability to make an action to cast a spell. Which is what I maintain that Spellcasting (generic ability) is just an action to cast a spell. And that action is not a
Special Ability.
Is making a standard action (to cast a spell) a
Special Ability? No it is not. If it were it could not be ex as it can be disrupted, nor it can be sp, su or na for other reasons. Hence it is not a
Special Ability.
Also note that
Special Attack: Spells use the same rules as the characters with few stated exceptions. So it means that the primary source for
Special Attack: Spells is in fact PHB.
So let's do an exercise and take our finely seasoned delicious king-fish the Aboleth Mage (MM1 p.8-9) to the table.
It has the following SQ and SA fields:
Special Attacks: Enslave, psionics, slime, spells
Special Qualities: Aquatic subtype, darkvision 60 ft., mucus cloud, summon familiar
How do we categorize these
unusual abilities? Ok, I'll try.
Special Attacks:
Enslave (Ex) - that's easy as the description has the type.
Special Ability indeed.
psionics (Ex) - easy, same as above
slime (Ex) - easy, same as above
spells - ouch much harder; I maintain that this is only about performing actions thus do not need a type at all. MM1 provides for a generic term
unusual ability.
Special Qualities:
Aquatic subtype - um, not a
Special Ability at all; Defensive unusual ability?
darkvision 60 ft. - um, same as above
mucus cloud (Ex) - easy as the description has the type.
Special Ability indeed.
summon familiar - huh, 24 hour action that uses 100 gp of magical materials (PHB p.54), not a
Special Ability at all and definitely not Ex as this action could easily be disrupted. Defensive unusual ability?
As you can see the categorization based solely on
Special Abilities is simply not possible nor a valid way to do it. The
Class Feature: Summon Familiar is yet another example where categorization with
Special Ability tags does not work.
So
Special Attack: Spells is not a
Special Ability. It is a
Class Feature that only enables specific action (to cast a spell).
I have shown with both MM1 and PHB rules quotes that
Special Attack: Spells is not extraordinary nor a
Special Ability. I have provided with rules quotes that Spellcasting is only about making actions and is not a
Special Ability.
TL;DR
There can be Special Attacks that are not
Special Abilities as per MM1 p.6 and PHB p.154.
The same applies for Special Qualities.
Special Abilities and Spells are two distinct terms that are separated from each other MM1 p.315.
Primary source for
Special Attack: Spells is PHB p.180 and for small portion MM1 p.315.
Special Attack: Spells cannot be extraordinary MM1 p.315.
Special Attack: Spells is not typed with
Special Ability tags MM1 p.315.
Class Feature: Spells cannot be extraordinary PHB p.24 and p.180.
More rules quotes.
Class Feature: Spells is not typed with
Special Ability tags PHB p.24.
Special Attack: Spells is governed by the same rules as
Class Feature: Spells MM1 p.315.
The generic ability to make an action to cast a spell is the Spellcasting-ability PHB p.24.
An action (standard, swift, etc.) is not a
Special Ability PHB p.138.
Hence
Special Attack: Spells is not a
Special Ability thus needs no type.