Newton's theory consists of many statements and definitions. It's arbitrary to cherry-pick part of it and call that "the core." We might as well say that God is at the core of his theory since God implicitly makes for the absoluteness of universal aspects of nature, and God keeps everything from collapsing.

The second law says how some factors relate to each other. To falsify it, you'd need the miracle of a massive object accelerating by way of no force. (Maybe an anti-gravity engine?) Newton is just defining force as that which accelerates a mass, accounting for the observed rate of change. If a body accelerates, he explains the motion by positing a force such as gravity.

These kinds of generalizations aren't easy to falsify, especially if taken in isolation, because they're fundamental to how the human brain thinks of nature.

But the medium was also the message with Newton; the point was the universality of his statements since he was tying together terrestrial and extraterrestrial motions. That universality could have been falsified by the ancient cosmologies, according to which there's a fundamental difference between the heavens and the earth. Newton's approach was thus an improvement on the older, more anthropocentric conceptions.