Yes. Merriam-Webster's dictionary says "specify" is "to name or state explicitly or in detail".
You could use it for anything someone clearly identified, whether or not you repeat the exact identification in your sentence:
The constitution specifies January 20 as inauguration day.
The date is specified explicitly in the constitution.
Our instructor specified single spacing for our book
He specified the spacing for our book reports.
The recipe does not specify the size of the eggs.
The recipe does not specify extra-large eggs.
But "specify" is not for things that are vague, general, very complex, or unknown, or for non-factual statements.
NO: He specified that he had no idea what happened.
NO: The patient specified that she felt awful.
NO: My brother then specified, "Owww, stop! You're you're killing me!"
NO: The candidate specified that the economic and social situation would have to be ameliorated in some way in order to extend the
American dream to all parts of the state and all demographic groups.
NO: Then my dad came in and specified, "What the heck are you doing?"