TL;DR: NO, at least not at my table.
Determining type of attack
You get a touch attack, which is defined as:
Touch Attacks: Some attacks completely disregard armor, including shields and natural armor—the aggressor need only touch a foe for such an attack to take full effect. In these cases, the attacker makes a touch attack roll (either ranged or melee) [...]
A touch attack roll is an attack roll against the touch AC:
Attack Roll: An attack roll represents your attempt to strike your opponent on your turn in a round. When you make an attack roll, you roll a d20 and add your attack bonus. (Other modifiers may also apply to this roll.) If your result equals or beats the target's Armor Class, you hit and deal damage.
Action type consumed usually
Up to here, nothing indicates the usage of an action. What is an action anyway?
Action: An action is a discrete measurement of time during a round of combat. Using abilities, casting spells, and making attacks all require actions to perform. There are a number of different kinds of actions, such as a standard action, move action, swift action, free action, and full-round action (see Combat).
Now, there is a pretty list in the combat chapter what actions are what type. An extract:
Attack (melee), provokes no AoO
Attack (ranged), provokes AoO
Attack (unarmed), provokes AoO [...]
cast a spell, provokes AoO [...]
So, here our touch attack needs to be typed. because it is part of a touch spell, it has to be (usually) an unarmed strike:
Touch: You must touch a creature or object to affect it. A touch-spell that deals damage can score a critical hit just as a weapon can. A touch spell threatens a critical hit on a natural roll of 20 and deals double damage on a successful critical hit. Some touch spells allow you to touch multiple targets. You can touch up to 6 willing targets as part of the casting, but all targets of the spell must be touched in the same round that you finish casting the spell. If the spell allows you to touch targets over multiple rounds, touching 6 creatures is a full-round action.
Intermediate Conclusion on type
So, without any rule changing parts (which come in later), the touch spell is an unarmed attack roll against the touch AC that occupies a standard action and provokes an AoO, while the spellcasting provokes an additional AoO because it is a separate standard action.
Changes to this behavior
With that in mind, we look at the quoted part that changes this standard behavior:
Touch spells in combat: Many spells have a range of touch. To use these spells, you cast the spell and then touch the subject. In the same round that you cast the spell, you may also touch (or attempt to touch) as a free action. You may take your move before casting the spell, after touching the target, or between casting the spell and touching the target. You can automatically touch one friend or use the spell on yourself, but to touch an opponent, you must succeed on an attack roll.
Touch Attacks: Touching an opponent with a touch spell is considered to be an armed attack and therefore does not provoke attacks of opportunity. The act of casting a spell, however, does provoke an attack of opportunity. Touch attacks come in two types: melee touch attacks and ranged touch attacks. You can score critical hits with either type of attack as long as the spell deals damage. Your opponent's AC against a touch attack does not include any armor bonus, shield bonus, or natural armor bonus. His size modifier, Dexterity modifier, and deflection bonus (if any) all apply as normal.
Conclusion on the action type
Ok, let's start with the changes: The standard action to do this special attack gets changed to be a free action and also the Unarmed attack gets changed to be considered an armed attack, no longer provoking an extra AoO.
Nothing here indicates you get more than one attack or that other touch attacks after missing (while still holding the charge) would be also a free action or considered an armed attack in contrast to an unarmed attack they usually are. That is, there is no notion of the number of these attacks with the exception of a Magus' Spellstrike ability about which, see below.
Conclusion, based on failure to indicate it is multiple attacks
Because of the failure to prove one might do more than one AoO-free touch attack, I interpret the sentence "In the same round that you cast the spell, you may also touch (or attempt to touch) as a free action." as "You get one free touch attack that is considered armed as part of the spellcasting".
This interpretation is further bolstered by the Magus' Spellstrike ability, that clearly states it is meant as only one armed touch attack that can be made with it, replacing the one touch attack gained from casting a touch spell in combat (emphasis mine):
Spellstrike (SU): At 2nd level, whenever a magus casts a spell with a range of “touch” from the magus spell list, he can deliver the spell through any weapon he is wielding as part of a melee attack. Instead of the free melee touch attack normally allowed to deliver the spell, a magus can make one free melee attack with his weapon (at his highest base attack bonus) as part of casting this spell. If successful, this melee attack deals its normal damage as well as the effects of the spell. If the magus makes this attack in concert with spell combat, this melee attack takes all the penalties accrued by spell combat melee attacks. This attack uses the weapon's critical range (20, 19–20, or 18–20 and modified by the keen weapon property or similar effects), but the spell effect only deals ×2 damage on a successful critical hit, while the weapon damage uses its own critical modifier.
Final conclusion: In case of emergency it is the GM's Choice
You may, of course, feel free to interpret it in the way of granting unlimited tries as free actions, but then you'll have to convince your GM that a guaranteed touch attack1 is not a game breaker and as the rule is intended to be. Under RAW either interpretation is possible (though unlimited attacks seem pretty far fetched), so your GM's Interpretation is what you have to inquire about. In this case, it boils down to (emphasis mine):
Free Action: Free actions consume a very small amount of time and effort. You can perform one or more free actions while taking another action normally. However, there are reasonable limits on what you can really do for free, as decided by the GM.
1Guaranteed hit argument:
Assuming somebody has an unlimited amount of retries on the attack roll, at some point one of those rerolls will be a 20. As a 20 is an automated success and having an unlimited amount of retries, the success of the touch attack would be guaranteed. Because of this, one could argue that you wouldn't even need to roll - defying the whole point in making an attack roll in the first place.