SOFTEN 'EM UP: After using the Throw a Grenade ability and hitting one or more targets, after actions resolve Master Chief may make a ranged combat attack targeting one of the hit characters as a free action.

Can this ability be used after any friendly character uses the "Throw a Grenade?

It would say have to say "after any friendly character" if it was to work that way.

As it is, this is MC's power so it only works when MC himself meets the requirements to activate it.