No! A lot of people seem to think this is okay, but it is not.
Bike helmets are designed to primarily prevent the front of the head from impact with a flat surface (the road).
Horseback riding helmets are designed to prevent this, but also to take impacts from a sharp object - such as a horseshoe.
Bike helmets, on the other hand, are better able to take skidding or sliding impacts. Also, bike helmets put more protection at the front of the head and riding helmets put more at the back.
In other words, no, you should not ride in a bike helmet.
You also should not cycle in a horseback riding helmet.