Electrolysis used to cause a Chemical Reaction?
You wouldn't need to supply electric energy to get oxygen and hydrogen to recombine into water, because the free energy change for that reaction is negative. In other words, the products have a lower potential energy than the reactants, and so the reaction will proceed "downhill" on its own. You would need to overcome the very small activation energy, which you could do with electricity or with a flame or spark. But once it got started, you would not need to supply additional energy, and could in fact generate electricity from the reaction.

This is the basis of how hydrogen fuel cells work - hydrogen and oxygen react in a device that is set up to extract the energy that is produced in the form of an electric current.

An easy way to think about it is that if you need to supply energy to make a reaction happen, then the reverse reaction will produce the same amount of energy (if we ignore losses due to inefficiency) or less (if we don't). The reverse is true as well.

On a side note - electrolysis specifically refers to splitting a compound. The "-lysis" suffix means "break apart," and "electro-" prefix means "with electricty." In your case, you are thinking of "electrochemistry" - which just means "using electricity with chemistry." Electrolysis is one thing you can do with electrochemistry.