Admissions - University - Graduate School. Is this a commitment?
Not knowing the policies of the university and whether the offer from the professor amounts to a commitment or not, I'd say yes, you can apply elsewhere, but as you have given at least a verbal/email/whatever assurance, you can't accept any other offer. 

I would make other inquiries somewhat discrete, however, so as not to cause concern should the first professor learn of them. You also shouldn't represent yourself as being entirely free to accept an offer in the interim. 

But you can also inquire of the administration of the first university what is the actual meaning of the offer given and accepted. Can you expect paperwork to follow or is it still tentative? You have a right to know this, of course. 

You could also, perhaps, ask this question of the professor involved, asking him/her whether you should keep looking or is it a signed-sealed-delivered offer. 

You don't need to leave yourself at risk, but you shouldn't put others at risk either. 

And congratulations.