could one say the Islamic Conquests were a response to Christian Expansion in the Middle East?

That's...implausible in the plain sense meaning of "response", and I have never come across anything suggesting that early Muslims thought of their conquests in these terms.

On the contrary, it doesn't seem to me that they could have thought of themselves in these terms given that in the time of the conquests 1. Islam seems to have been an exclusively Arab religion, in that you had to make yourself a client of a tribe in order to convert, 2. it's not entirely clear that at this point the new faith conceived of itself as a full break from Christianity. While the Quran discusses "people of the book", Fred Donner posits that early Islam was an ecumenical "believers movement" of monotheists, of which early Muslims would not have viewed themselves as distinct. Among other examples, he and others have noted the use by the conquerors of Christian churches for prayer that were not then converted into mosques, in a way that strongly suggests that they simply shared rituals.

would the Khalif have seen them as a potential threat? I am assuming this because of the sheer size of the two Empires (and their past expansions.), and both have rapidly expanded in the past,

That strikes me as unlikely. Neither empire had ever really bothered with conquering this territory. It was far easier to simply buy them off, as with the Lakhmids and Ghassanids. And again, I'm not familiar with anything suggesting that early Muslims were concerned about this.

on top of that the Romans probably saw the Muslims as infidels/heretics, and might attack the Khalifa.

/u/Shlin28 has a great post here on Byzantine attitudes towards early Islam. To summarize, there isn't really any evidence that the Byzantines viewed Islam as a heresy. They were more likely to describe it as related to Judaism or as some kind of proto-Abrahamic faith. Or even more, likely that they simply ignored it because the actual heretics within the empire were viewed as a far more pressing matter.

He quotes Robert Hoyland:

'his [Muhammad’s] religion was nothing new, indeed that it was primitive, not having benefited from any of Jesus' modernisations'. Christians therefore understood Islam as many things, perhaps even as something they despised, but it was not a Christian 'heresy'.

So why did the Arabs expand? Well, we can't know for sure what Muhammad's motivations were, but the precedent of expansion beyond Arabia had already been set during Muhammad's lifetime, with an expedition into Syria in 626.

This further complicates the "response" idea, as in 626 this territory was still under the control of the Sassanians.

While we can't say definitively, it seems highly plausible that the routes of expansion followed 1. pre-existing trade routes, and 2. pushed into territories where Arabs were already residing.

I wouldn't discount religious or even apocalyptic fervor as a motivation for conquest, and indeed if the apocalyptical literature is anything to go by that was clearly present, but it seems more likely to me that the conquests simply snowballed until they met points of final resistance or were incapable of expanding further, rather than carrying out some sort of policy of preventative security.