Someone may have a better answer than me but I would say at minimum you should give yourself 2 hours. The roads are winding and you won't be able (or shouldn't go very fast). There are several overlooks and places to hike throughout so that's just a minimum in my opinion.
At the speed limit the park rangers told me it takes 45 minutes to get to the far edge of the park where most of the cliff dwellings are. I spent 4-5 hours between driving, short hikes and general exploring.
It only takes a few hours to just drive thru but I recommend doing some of the tours, like Cliff Palace. They ate worth it!
Mesa Verde isn't quite one of those places you can see everything quick. There are multiple overlooks, cliff dwellings, and hikes. But there is a fork in the road and to do each one is close to an hour down and an hour back. But it's worth the time it takes.
At least 3hours
To drive to some overlooks take between 2 and 4 hours
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