Not quite.

I wish I ate meat indicates that, for whatever reason, she does not eat meat. Technically, this could be by choice, it could be for medical reasons, etc.

I wish I could eat meat indicates that she is prevented from eating meat, whether she chooses to or not. Possibly her religion forbids it, or she has a health problem that prevents it, either way there is some external factor or force that prevents her from eating meat.

Most likely from the context, she is vegetarian by choice, as this is a common phrase when someone is abstaining. Another example would be "I don't drink" to indicate abstaining from alcohol.

Technically, the first phrase doesn't indicate the reason she does not eat meat; it simply states that she doesn't. The second phrase indicates both that she does not eat meat, but also indicates that there's a factor or force preventing it. Without that indication, the assumption is that it is by choice.

To add on to Werrf's correct answer: This subtle nuance is reflected in many English sentences. For example, suppose you are at a party and offer someone a glass of wine. They respond:

I don't drink (alcohol).

The simple present refers to things that are general, natural, or habitual, so this expression means the person doesn't drink as a habit. We generally assume this is due to personal or religious reasons, rather than medical reasons, but we'd have to ask to know for sure.

In addition, it's natural to assume this is a relatively long-term habit, rather than something recently adopted. The following conversation would be odd, and possibly funny:

A: Hi, would you like a glass of wine?
B: Sorry, I don't drink.
A: I understand. When did you stop drinking?
B: Yesterday.

Compare this with:

I can't drink (alcohol)

This reflects a general inability rather than a habitual practice, and also indicates a contrast with the person's desire -- they would really like to drink, but for some reason they're not allowed. Otherwise, if they didn't want to drink, they would say it the other way, "I don't drink".

In addition, it's likely to be a short-term rather than a long-term inability, but (depending on the context) it may be impolite to assume.

A: Hey, we're all going out after work to get a beer. Want to come?
B: Sorry, I can't drink.
A: Well, perhaps tomorrow then?
B: No, I mean I can't ever. I'm allergic.
A: Er ... sorry about that. You can come with us and order something else, if you want?

This scenario may tell a lot about the sentence in question. Considering it's about a man cooking for his girlfriend and she says "I wish I ate meat" after being presented with the meal, I would assume she is saying "I wish I ate meat" in the context that she thinks the meal looks good but sadly she does not eat meat so cannot partake in that portion of the meal. I am guessing there would be some emotion expressed on the person's face and some inflection in their voice to convey this underlying meaning. If you could say what movie I could be more specific about it.

If my assumption about the emotion of the scenario is correct then both sentences would mean roughly the same thing.

"I wish I ate meat". would convey a choice not to eat meat. Like a vegan.

"I wish I could eat meat". would mean more like they cannot for whatever reason eat the meat. Some religions prevent people from eating meat and that would fit here well. Maybe the choice of the parents restricting her from eating meat. Possible other reason.

In both sentences in this scenario I believe both would convey a desire to want to eat the meat but sadly they cannot. Meant as a complement to the cook.

Update:

In response to your updated question and comment below here is what I have to say about this sentence.

@JoelVermish I have not watched that movie but I have seen the trailers for it. That is what we would call a Romantic Comedy AKA "Chick Flick" so I am sure there is a lot of emotion in that movie. That said, I believe it was also meant to be comedic as it is about a women trying to get rid of a guy in 10 days (I think for a book) but instead she falls in love with the person over time. In that case she may also be saying "I wish I eat meat" sarcastically in an attempt to off put the man so he becomes disinterested in her as part of her ploy to get rid of him in 10 days.