"Non sai se ha un cane."
It's highly unlikely that this sentence would be used in isolation - so you would be able to tell who the subject of the verb 'ha' is by the context (ie. by the earlier part of the conversation or by gestures/body language). If the subject can be deduced from the context, pronouns are not used - unless you specifically want to place the emphasis of that sentence on the subject.
Since the verb carries the person and number, subjects are often left out in Italian. In facts this sentence sounds natural.