No. This is a post-Cold War myth, written largely by people who would like to make SDI not appear to be the boondoggle it was, or to make it look like something that had a positive effect on diplomacy, rather than the more easily-documentable negative effect. Pavel Podvig has written at length about this here and here.

Aside from being asserted without evidence, I would just note that the fall of the USSR was clearly caused by many factors, most of them internal: Gorbachev's attempts at opening up the system very clearly and directly led to its instability and failure. While the Soviet overexpenditure on arms (in general) no doubt did not help its overall economy, the idea that its collapse can be traced to that, much less to a specific US program which the Soviets did not in fact respond to, is facile.

I'll shamelessly plug an answer to this very question that I wrote, but basically....what u/restricteddata said.