No magic means there are no arcane spellcasters at all? and no divine casters too?
Or just no magic items?
If this is your first time DMing, I heavily advise against any houserules, especially such drastic ones, that change the entire game balance.
You need to get a decent understanding of the balances and mechanics before tinkering with them. And this is not something DMs can describe to you on a forum... it's something you need to know from experience.
If this is not your first time:
Make a list of available classes, don't go "you could, I advise against it".
What's wrong with paladins and monks when magical classes are not available? because otherwise there's fighter/barbarian/rogue/ranger?
Also, how do you intend to handle the healing? Be aware that spellcasting draining HP means somebody patching people back together. Without magic?
What's stopping some PC to hand his gems to another PC because he doesn't want spells? What will be the consequences?
Honestly all of this sounds like too big a change... I seriously advise against it. I fear this will end with a martial-centric group who'll just ignore spells altogether as in a no-magic world enemies shouldn't have access to magic either.
If you're saying no to something in your game, say no. Also restrain yourself from saying "you players get no magic..." and then outfitting your NPCs with magic. That being said, your gems sound cool. If someone has 2 of them, can they combine them for interesting effects?
Fire + Healing to give the effect of Infernal Healing or Regeneration or something?
Also unlimited casting/health: I say take a look at the old Dark Sun game. They had something similar. Also there's Runequest where everyone had a stat called Power. This stat (and supporting storage battery items) were the reservoir casters could dip into to cast spells.
Frankly I think that's more reasonable. If all spells are divine, they're powered by Wisdom. Everyone has a number of Power points = to their Level and Wis mod. Once these are evaporated, the caster can keep casting, but they take Wis damage and risk permanent madness.
Then give every gem a set amount of spells, and have these spells require Power = to level or something. If you find a gem of Wish, you can have unlimited wishes, but each one drains say 9 Power. If you're a 20th level fighter focused on Wisdom, you can have say 3 wishes in the day and take 1 pt of Wis damage, but you'd be ok. However if you're a 1st level commoner you'd use this gem once and go brain dead.
I get the impression that you've got this neat idea for gem-magic that anyone can use and you're removing traditional magic so that gem-gamic shines. I might suggest (as Kyoni) that you not make such sweeping changes to the system your first time out. Instead, let you players know that you want to experiment with a tacked-on system that may or may not need some balance changes after launch.
Let your players play whatever they want. Traditional wizards, clerics, whatever. But those who decide they want to play straight martials like fighters or barbarians know that there's this interesting gem-magic thing that they can dabble in.
Test your new idea but keep the fundamental game intact so you know if your system is even vaguely balanced. Once you build such a system, then you could reasonably consider relying on it.
Just my thoughts.