No.
Short of physical invasion, what brings about any liberal democracies demise is internal, not external. The fact is that there are large swaths of the planet that do not agree with Liberalism and have no intention of adopting it in the near future. Most of sub-Saharan Africa is communalistic, not liberal. The same could be said for Asia.
Your argument is similar to the argument that the Soviet Union would be the demise of liberal democracies back in the 1950’s and 60’s. That the Marxist ideals were so incompatible with Liberal Democracy and capitalism that they could never be reconciled was common back then.
In fact, I would argue that the decadence of liberalism is more a threat to Islam than Islam is to liberalism. Women are driving in Saudi Arabia and Iran, there are two McDonalds in Mecca, and Justin Bieber plays concerts in the UAE.
Islam is unique in that it is still largely a religion and a political system at the same time. But so was Judaism. Arguably, so was Christianity from late Roman times until the demise of the Holy Roman Empire.
No, if Western Liberalism fails it will not be because of Islam, or any other outside source. It will be because its people, being so steeped in the individualistic ideal of Liberalism, will no longer feel compelled to fulfill any of their civic duties. It is the curse of Liberalism – catastrophic success.
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