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"Its" is usually used as an adjective. However, I sometimes encounter it being used as a pronoun, as in "I ate my dinner and the dog ate its." Is this an accepted use of "its"?
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This utterance sounds a bit odd to me. You usually refer to a person or animal you are not familiar with by means of the impersonal "it". Of course, you will never say:

* I ate my dinner and the strange man ate its.

In such a case, "his" is the correct form.

However, when you talk about an animal and pay attention to it for any specific reason, noticing if it is a he or a she, you may say:

I ate my dinner and the dog ate his (or hers).

So, in my humble opinion, it should go like this:

I ate my dinner and the dog ate its meal.

What if it was its breakfast? Emotion: wink
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Giving more information on the word "its" "Its" is either used as an adjective or a personal pronoun. It is written in third person possesive case. Which means it shows possession. It all depends on how you use it in your writing. Some people get confused with the word it's thinking that it's shows possession because of the apostrophe. Technically it's is a conjuction of it is. To learn more about the word "its" I suggest either you reserch it on the internet or ask your english teacher.