Different editions have gone different ways, but in 3.5, you don't lose prepared spells (or unused slots, for spontaneous casters). Unconscious characters can't concentrate, which may end some spell effects, but they keep any spell slots that were available to them.
That said, it is worth noting that not all editions work this way. In 2e, for example, you do lose your memorized* spells—they're "forgotten"—when you drop to zero (though 2e also had different rules concerning when you actually die). This has sometimes been a source of confusion among people who have played both.
* In 2e you "memorize" spells instead of "prepare" them, which is mechanically equivalent though slightly different fictionally.
Can the GM ban references?
Yes. Specifically, "the D&D rules help you and the other players have a good time, but the rules aren't in charge. You're the DM, and you are in charge of the game." (DMG p.4, "The Dungeon Master," emphasis in original.)
That being said...
This is strange, as presented. In nigh-thirty years playing I've not run across a GM who disallowed looking up references. Not during your turn? Sure. Not at all? Weird.
(When running a table that operates under time constraints I will often skip a player who's not "ready" for their turn in combat. This tends to fall harder on spellcasters than on martial types, and harder on new players than on experienced hands. Of course, this is articulated at the beginning as a standard of play.)
But according to your edit, that's not what's going on here.
Your GM seems to be basing this restriction on two ideas: the character needs some time to be able to "fully" use their class features, and the player's skill should impact the character's abilities.
The character can't use their class features--new spells, in this case--until they're "broken in."
This has no basis in the rules. In short, "Beyond 1st level" (PHB p.15) tells us that when your character earns certain XP they gain a level. The class descriptions tell us what new things the character can do when they gain a level.
@Kryan's answer has the right of this: the two new spells your wizard now knows by dint of increasing a level represent the work that character's already done to learn new spells, not some new task the character needs to take on.
(As an aside, do fighters only get part of their Ability Score Improvement or feat until they've sufficiently proven themselves at their new level? Do druids pop out of Wild Shape suddenly because they're not yet well-trained in a new form?)
Where there is some support for something like this is in the question of when a character earns XP or can gain a level. Adventurer's League rules, for instance, only allow a character to gain a level when they've completed a long rest or at the end of a module. That is, if killing goblin three of seven in an encounter would put you at a new level, we usually don't stop combat to do it then.
This is also subtly achieved by GMs who award XP at the end of sessions, or by "milestoning."
Again, though, once you've got the level, you get all the class features that come with it, full stop.
Player skill = character ability. Charging your character an action for you to look at your character's spells is... insane. That's a huge hit in the action economy.
This seems to stem from an idea that the player's ability to memorize everything redounds to the character's ability to perform in the fiction.
The idea that player skill should be important is an old one, and has plenty of merit to it, I think.
But this is an incredibly ham-handed way to bring player skill into the game.
(Again, I've got to wonder if your GM tries to stab the fighter-player just to see how well the player reflects DEX 18?)
"Shouldn't I be allowed to reference my spells...?"
This is hard to answer: there may be things going on that we're not aware of. What you should be able to do, without question, is talk to your GM about what's going on. Ask them what purpose they see their rule serving. Ask what they're hoping to achieve. Describe the difficulty it's causing you. Ask them for help playing your character well at their table.
(This all assumes the very best of your GM: that there's a good reason, poorly articulated, for this and that they're interested in helping you play your character. I hope that's your situation, rather than the other one: you're sitting at a table with a petty tyrant who hates spellcasters and is actively trying to make play difficult for their players. In that case I suggest you find a different table.)
Best Answer
No, they don't lose spells
There are no rules about losing known spells when unconscious. Doing so may seriously upset the balance of the game, but always discuss house-rules with your table if it's something you're interested in.
Concentration
The Concentration mechanic for spellcasting may be what you're looking for. This rule does state that any spells that require Concentration for ongoing effects end if the caster becomes Unconscious.
Considerations
In 5e, spellcasters either have a list of known spells (and generally can only swap a spell for another when they level), full access to the list (but prepare known spells daily), or have known spells that they can prepare daily. Losing a spell would be a massive ongoing hindrance, as well as difficult to track what is known/gone both on a short term and long term basis.