Well "large" is a debatable term, but the military spending certainly had an impact, however in my opinion there are various inter-connections and active dynamics that play a larger role than just the military spending. What I'm going to do is list a few things and explain how they acted upon each other even as they individually contributed to the Soviet collapse. We'll start with the military spending.

• Military spending as a thrown gauntlet - I want to preface this by admitting I'm not really a fan of Reagan's presidency, but here we go. Reagan continued and then ramped up military spending/build up that had actually started near the end of Carter's term. The Cold War was over 30 years old at this point, and the Americans and Soviets had a fair understanding of one another's capabilities. The thing is, the U.S. showed during this build up that if it actively set aside resources and focused on military spending, it wasn't really an arms race but an embarrassment for the Soviets. Not only did the U.S. show for the final time that it could re-tile, mop, and then sweep the floor with the Soviets in any kind of military build-up race, it showed (partly in truth and partly due to the Soviets simply not understanding that Reagan was off his rocker), that the Americans could come up with weapons and weapons' systems to which the Soviets had no answer. Not only did they have no answer for such weapons as the Star Wars system, but they understood they had no way of coming up with such a system in the near future. This ties in to my next point.

• Computers/ARPA/DARPAnet:
  Here we have a connected factor. The Americans and Western world not only had far better computing technology, the level of integration simply dwarfed the Soviets. This is not only from a military perspective, but on a consumer/business level the Soviet Union was completely unwilling and unable to allow its citizens the level of access to both computer technology and the internet that the Western world saw as a convenient necessity. (I think, I'm sure of the comparison but not of the source) D.P. Nickles has something in Under The Wire about the almost laughable disparity between the two sides when it came to computers and the internet, but points out this this was also by Soviet design, as Soviet officials actually wanted to keep the Western world out as much as they could. This leads to...

• Helsinki Accords/Culture Cracks the Curtain: To make a very long and interesting story short, Western Europe and many of the Soviet satellites in the late 60s and early 70s had basically had enough of the U.S. and U.S.S.R. "ruling" the world and trampling on rights when they perceived the other to be active. The Helsinki Accords, which to this day boggle the mind of anyone who dares consider why the Soviets actually signed on to them, essentially guaranteed a certain level of human rights and freedom to the citizens of the Soviet sphere. This meant that a variety of factors which the Soviets had tried to keep out, now seeped in at an ever increasing rate. This leads to my penultimate point, which is a kind of amalgam.

• Nixon/Detente/Gorby/The World: Few would deny that Nixon's Detente (though many would argue whether Nixon originated the idea - not important here but I'd say he'd been around the barn long enough to know which smells came from which animal), Detente, Gorbachev, and other "splitting" factors contributed to the Soviet collapse. Another long story made short - Nixon not only heightened the tensions between the Soviets and Chinese, he also enhanced the already active slide both states were experiencing (one intentionally the other not so much) towards more integration rather than isolation with/from the world. While detente was effectively abandoned, the tumble down the hill had already started. This meant that while Brezhnev would continue to be crazy, others of influence who would rise (Gorby) would see a different kind of challenge. Gorby essentially thought he could loosen up the U.S.S.R.'s hold and make it a stronger state by slowly letting in capitalism (quite similar to the China model). However, where China was more or less a cohesive society before Mao or Communism came to power, the U.S.S.R. was hardly stable at any point in its history. Gorby loosened the reins (which a certain Vlad took and still takes issue with) and the horses bolted. Gorbachev sincerely believed the Soviet Union could and should keep going, but the world, and the peoples in the Soviet satellites thought otherwise. On to my final point:

• C-c-c-combo:
  All of these factors, plus others (Vietnam, the "third world," the Afghanistan invasion/occupation/bleeding), contributed to the Soviet collapse. But they also influenced and shaped one another. The West (plus "western aligned" Asia) was becoming more socially, culturally, and certainly economically integrated (some...like Walter LaFeber, might argue dominated). Reagan, Thatcher, and their heavy-handed and rhetoric rich styles were not something the Soviets could not, in their prime, have dealt with. However, the Soviet Union had been slowly decaying in almost every sense (special exception being bio and chemical warfare, since, although the Soviets had agreed with the Americans to stop all production and experimentation in this, the Soviets never believed the Americans had actually stopped [which they had] so they kept going right up until the collapse). So, when Reagan started pressuring the Soviets to go another round all out, like they'd been able to do in the 50s and 60s, the very problems the Americans always predicted would happen to the Soviets finally showed through. This might not have been a death knell, but by this point, Western Europe was incredibly integrated and while not wholly at peace was on the best terms it had been in its history, the U.S. had such economic and technological superiority that it was clear to anyone with a brain that they'd already won.

So, significant, yes, but significant as part of a dynamic system of influences, any one of which the Soviets might have been able to deal with but which as a combined and continuing rolling wave of influence simply knocked over a hollow and rotten husk.