Unfortunately the plumber didn’t tighten it

Maybe the plumber should come back.

^^^^ I wouldn't want to try to tighten it myself + I also wouldn't want a gap or to fill the gap with silicone caulking. The plumber should have done it correctly. I'd request a call back to finish the job.

To tighten the hot water heater needs to be removed. He won’t do this for us.

You cannot seal or effectively caulk that. The plumber messed it up the plumber needs to fix it.

Exactly who is the 'we' that installed the faucet? Homeowner or plumber?

Shouldn't there be some sort of gasket between the fixture and counter top?