Yes, it can. The speaker did feel bad, and the speaker did eat too much. So it makes sense and is grammatical.
However, you seem to prefer had eaten. I agree with you. Had eaten tells us that the speaker finished eating first, then felt bad. So it implies a cause and effect. To me, that's a more interesting sentence.
On the other hand, Ernest Hemingway would have disagreed. He made his fortune with lines like: "I ate lunch. I ate too much. I felt bad. The wine was red. The hills were brown."
So it's mostly a matter of taste. But I think you might be a better writer than old Papa Hemingway....
[I edit writing and have tutored college writing.]
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