Bush appointed Osborn, but Obama approved him.
Based on the reality that the Bush Administration provided a smooth transition for the Obama Administration, former President Bush would have not done something as drastic as disbanding S.H.I.E.L.D., setting up H.A.M.M.E.R., and appointing Norman Osborn without Obama giving the okay to go forward with it. And considering that H.A.M.M.E.R. and Norman Osborn are under the full jurisdiction of the United States government unlike S.H.I.E.L.D., Obama could have easily undid what Bush Administration did on account that H.A.M.M.E.R. was not fully ready by the time the Obama Administration took office. And for that reason, former President Bush would not have done something that drastic without President-elect Obama's approval.
While President Obama did have reservations about the amount of power that Osborn obtained and because of certain allegations about Osborn donning the Green Goblin costume in Thunderbolts Mountain killing several C.S.A. agents, most of them were put to rest in Thunderbolts #129, giving him his full approval while expressing that Osborn has a tough act to follow by succeeding S.H.I.E.L.D.
So to sum it all up, Bush paved the way for Osborn's rise to power, but Obama gave it his stamp of approval, making them both equally responsible.