"I eat meat."
In a way, yes, but one doesn't say "itheann mé", it is always (as far as I know) contracted to "ithim". The only verb I hear not contracted in the first person singular like that is "tá". That is, one often hears "tá mé", and I believe it is considered acceptable use, though perhaps someone more knowledgeable that me should varify that before you use it.
I'm pretty certain that it doesn't have to be contracted every time to 'ithim'. In the short time that I have been doing Irish on here there has been at least two other instances where the first person singular has used the longer form.
Hopefully the marking will get more consistent...
Tá is really the only time it is used like that, in standard Irish. You'll hear it separated in Connacht Irish, but it's not standard Irish at all.