Can a Canadian prime minister refuse to swear a oath of obedience to the Queen of England ?

Yes. Can’t force someone to say something they don’t believe in. Although in the UK people are getting worked up about Jeremy Corbyn, leader of the opposition who is a republican and refuses to have anything to do with the royals. I have heard the Queen doesn’t mind. She’s not a stickler about what her subjects do, although she has high standards for her family performing their royal duties

Canada is an independent sovereign nation, so I would imagine so.