Can I ride a Trek Crockett (CX/gravel bike) in a road race if I have a road wheelset and tires on it?
Yes definitely, I raced all of last years criterium season on my Ridley X-bow (rim brake) with slick tires. 

The thing you may also want to consider is your gearing, if you have a 1x you may want to use a different cassette. 

Also if you your bike is disc brake equipped you may not be able to use it in some races. If you are racing in the UK then your in luck as British Cycling have recently announced there allowing of disc brakes in 2018 for road racing and crit, if you are racing somewhere else you will have to check your local stance on disk brakes.