Not even slightly. For one thing, Alexandria is most definitely not in the West, and wasn't in the business of storing up "western" ideas: there's no special likelihood that it even contained very much in the way of Latin texts. Secondly, it may have been the largest library in the Hellenophone world in the 1st century BCE, but it was far from the only one.

Of the threads in the FAQ that /u/Lomedae links to, this is probably the most pertinent one.

The analogy I usually draw is this: imagine that the Library of Congress and the British Library were to burn down tomorrow. Would the majority of human knowledge be lost? Of course not! And for exactly the same reasons: they may be the biggest centralised repositories of knowledge in the world, but they're hardly the only ones.

IMO the transition from the scroll to the codex is a far more damaging factor in the loss of Greek texts. And that didn't really get under way for another 200 years afterwards.

In addition to all that, /u/wedgeomatic's answer to this question is also worth quoting:

The loss of the library was so devastating that no one bothered to definitively record when it was destroyed and Alexandria was only able to remain one of the premier intellectual hotbeds of the Mediterranean world for the next 700 years.

Please look at these links to a some of the previous questions about the Library of Alexandria.

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