This is very unlikely to be true because the expansion of Rome and the decline of the slave population occurred centuries earlier than the Empire's collapse.

The only major territorial expansions of the imperial period after Augustus were Dacia and Britain, and these conquests both occured during the early Empire. The slave population also began to decline in this period, I believe mostly because of the lack of conquest. This would have changed the Roman economy, but if it drastically weakened it, we would have seen the effects around this time. Instead, the Roman Empire survived much longer, even overcoming other major economic disruptions, such as the raiding and banditry of the third century. It would not be until the fifth century that the Roman Empire collapsed.

Furthermore, in the fifth century it was only the western Roman Empire which collapsed. The eastern Empire survived for many more centuries even though it had also moved away from an economy that heavily relied upon slaves.

Economics definitely played a big role in the collapse of the Roman Empire, but it was not because of a lack of slaves. The western Roman economy suffered heavily when invading barbarian forces occupied both Spain and North Africa, denying the resources from those regions to the Roman armies. Conversely, the economic heartland of the eastern Empire (Anatolia and Egypt) remained largely under imperial control.