[RPG] How many rays can a beholder use in one full-attack action

Short answer:

Standard attack = 1 attack action = 1 attack roll at a listed bonus.

Full attack = multiple attack actions = listed number of attack rolls at the listed bonuses.

Now, let me explain that:

Quote from Pathfinder SRD:

Multiple Attacks

A character who can make more than one attack per round must use the full-attack action (see Full-Round Actions) in order to get more than one attack.

Therefore, if the creature in question had to move and only has 1 standard action to spend on attacking, which gives it 1 attack of its choice (as in, 1 attack roll - be it 1 bite attack, 1 claw attack or 1 gore attack).

In case of full attack, it would be able to use its full potential, attacking with 2 bites, 4 claws and 1 gore with listed attack bonuses and damage:

Attacks made with natural weapons, such as claws and bites, are melee attacks that can be made against any creature within your reach (usually 5 feet). These attacks are made using your full attack bonus and deal an amount of damage that depends on their type (plus your Strength modifier, as normal). You do not receive additional natural attacks for a high base attack bonus. Instead, you receive additional attack rolls for multiple limb and body parts capable of making the attack (as noted by the race or ability that grants the attacks).

Cue obligatory "The DM is always right. However..."

However, yes, there is such a rule. It's the very first sentence of the "Full Attack" section of the Combat chapter. http://www.d20srd.org/srd/combat/actionsInCombat.htm#fullAttack

If you get more than one attack per round because your base attack bonus is high enough, because you fight with two weapons or a double weapon or for some special reason you must use a full-round action to get your additional attacks. You do not need to specify the targets of your attacks ahead of time. You can see how the earlier attacks turn out before assigning the later ones.

emphasis mine.

You have haste, which adds one attack per round, plus you have the ability to make at least one attack baseline. That adds up to two, which is "more than one", so the very sentence your GM is citing in his reasoning is the explicit statement you're looking for to counter him.