What's the difference between using と and を in the sentence below:
They both mean, He wrote "eat" on the paper
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The latter is wrong. You should use quotative-と because 食べる is what is concretely written on paper. ～を書く is okay when the object is something abstract such as 文字, 漢字, 手紙 and 物語.
EDIT: Actually, as @broccoliforest pointed out,
「食べる」を書く is also acceptable in a rare context where the "message" or "content" is not relevant. If you say
「食べる」を書く, it sounds like the focus is on the appearance of the characters rather than the message conveyed by 食べる. For example, you may hear this when you are talking with your calligraphy teacher about how to write 食べる correctly with a brush. Although
「食べる」を書く is unrealistic in practice, you may safely say something like this:
I wrote the hiragana 'ME' 100 times for practice.