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If you mean, 'A fish also has a heart', then yes, grammatically it is correct.
But what posters are telling you is, it doesn't make sense in the context ( your reply to your girlfriend) in which you are saying these words.
The meaning you are wishing to convey is not that, you have a heart, I have a heart, a dog has a heart, and a frog also has a heart.
She says I have no heart. Even the smallest and lowest of creatures have hearts, so of course, as a human, I too have a heart/I also have a heart/I have a heart also.
But : "She says I have no heart. A fish also has a heart." doesn't make sense!