I turned back to the dog and continued watching.
Can I write the sentence like this or do I have to add "it" to make it grammatically correct?
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It depends on the context around the sentence. If you were watching the dog before an interruption, then the "it" will generally be understood by readers, and you can just say "continued watching." If you had other things that you were watching, then give the dog a pronoun ("watching it/her/him").