The best description of Newton's conundrum, and its modern resolution, may be found on pages 295-297 in the 1997 Basic Books ed. of Guth's book titled "The Inflationary Universe", which uses simple algebra and one diagram to illustrate that resolution.
Newton's thinking (accurately described in John Hunter's answer) contained one flaw: He failed to consider the possibility that the collapse of even an infinite universe might occur "everywhere at once", in a reversal of the content in the phrase (much-used in modern classrooms) about our universe's "Big Bang happening everywhere at once". To use Guth's exact terminology about an observer in Newtonian space, "No matter where the observer might be in the infinite space, he would see all the rest of the matter in the universe converging towards him".
Of course, what we see through our telescopes is, happily, the physical reverse of that possibility: The observable region of our universe is expanding, away from us and away from everything else in it. That's usually presumed to be the case outside our observable region, under the assumption that the universe (whether it's a single universe, or a "local universe", causally-separated from other LU's basically similar to it, in an inflationary multiverse) is isotropic and homogeneous.
There are some more local exceptions, concerning astronomical bodies large enough for their gravitational attraction to overcome the expansion (whose main origin is most commonly supposed to be "Dark Energy"): For instance, when people refer to the fact that the Milky Way Galaxy will eventually collide with the Andromeda Galaxy, they're referring to one of those exceptions. More extremely "local" exceptions occur at microscopic scales: Dark energy has not been observed to tear molecules apart.