The Soviet Union was already heading down the tubes. Reagan could have sat in his office like a senile old man and watch the USSR crumble.
NOTE: I'm not making this up as I go along, this is essentialy Kissinger's analysis. The Soviet Union was so fundamentaly unsound by this point that it was going to collapse anyhow. Reagan simply made it happen a bit faster because Gorby was rather pathetic in foreign policy and leaped at the chance to get foreign support as his domestic support was crumbling.
According to Kissinger, Reagan showed little understanding of foreign policy and relied mostly of his apparent naivite and good will to cajole foreign leaders. If Gorby hadn't been quite so desperate for support and was more like Brezhnev, Kruschev, or (God help us) Stalin: Reagan would have done jack and the world would have known it.
I've said this before, and I'll say it again: the Soviet Union and other Warsaw Pact states were, for the most part, liberalizing. We all know about Walesa and Solidarity, movements like that were not uncommon. Reformists had already been elected in most of Eastern Europe under a combination of Gladnost/Perestroika and Gorby's rejection of the Brezhnev Doctrine.
Reagan had little, if anything to do with Gladnost/Perestroika as both were key components of Gorby's ideology. Gorby was no Communist hard-liner. He was very much Social Democrat in his outlook.