Could be. Depending on where you live (or for which country you ask), of course.
For Germany, first of all, a cyclist is forbidden to ride on a pedestrian path. But, on the one hand, there are mixed pedestrian/cycle paths, and on the other, this prohibition doesn't chage his rights with respect to other vehicles.
But since you ask about "crossing the road", then I would (over here) assume you are talking about a crosswalk. Now, there are crosswalks and there are crosswalks. A lot of them are regulated by traffic lights, but then the issue doesn't occur since the traffic lights regulate priority.
So a crosswalk without traffic lights, or "Zebra walk", where pedestrians per default have priority over the vehicle traffic on the road. This is where it becomes interesting. If the cyclist is riding the bike, then he's also a vehicle. As such, he's not permitted on the pedestrian path, and if he chooses to cross the road at such a crosswalk, has no specific rights - same as if he would cross the road at any other place.
If, however, the cyclist gets off and pushes his bike, then he becomes a pedestrian and as such would have priority in this special case.
Yes, German trafffic law for cyclists is complicated - don't even start the discussion about which traffic lights a cyclist has to obey in which situation.