There is a long list of plausible causes of stagnation of Roman Empire but the prevalence of slave economy was likely only an indirect one. I am more inclined towards the social one — the dominance of landed aristocracy vs merchant classes/intellectuals. We saw similar dynamic in action in Europe as well (England vs Spain for example) — that is probably the main reason why industrial revolution took off where it did. If people thought that pursuit of profit, of innovation, was something that society valued they did it with vehemence (and still do in places where this dynamic exists — say Silicon Valley or today’s China). In places where it was looked down upon (most of feudal Europe, old China, even say ancient Egypt for completeness sake) they did not. Another major reason which probably was decisive in the spread of the innovation based growth was the fragmentation of Europe and ensuing competition between states — something that did not exist in unitary empires like Rome or China.
Of course there is a much longer list behind it (security of property rights comes to mind — something pretty weak in any absolute imperial system), but slave based economic system probably wasn’t that significant. For starters slaves were not free in terms of cost, there was real upfront investment behind, ongoing costs involved and probably there were events where these costs went up rapidly (say any major epidemic, infrequent supply etc) that in any normal economy would lead to labor saving innovation. In Rome they did not. It is also good to remember that in most of countries in a Europe up to 19th century most population was in effect indentured — as feudal peasants (often restricted from leaving their villages) so at least in agriculture the system wasn’t that much different. The indenture/slavery based systems did however enhance role of landed aristocracy which was more conservative and less open to innovation (it also prevented movement of labor to industry). In this way, slavery based economic system did weigh down on progress of Roman Empire but probably no more than say feudal exploitation of French peasantry weighted down on industrial progress of pre-revolutionary France (which it did vs England for example).