Will the gravity of two non-ionic tungsten atoms at rest a hundred light years apart (many relevant details omitted) eventually draw them together?

The original question was evidently far too long to fit into the title, so I’ll post it here in the body:

"If two non-ionic tungsten atoms spontaneously popped into existence at rest a hundred light years apart in the approximate center of a hypothetical non-expanding universe that is a billion light years across but utterly devoid of all other ordinary matter and energy, including "dark matter," would the gravity of the widely separated atoms eventually bring them together into a close encounter or even an outright collision? If so, how closely will each atom approach the speed of light just before that encounter?"

I honestly have wondered about this on occasion for the longest time (in truth, decades) and hope that my obvious ignorance of physics hasn’t rendered the question as phrased trivial or otherwise uninteresting. I’ve tried to phrase it for maximum conceptual simplicity. Nor is it quite clear how to properly formulate this question for a straightforward Google query — my attempts thus far have attracted only a flood of irrelevant results that dance around the core question. The major search engines don’t seem able to cope with so many common physics keywords. -_-

P.S. No, this is not a homework question! I’m just an uneducated dude with odd questions buzzing around in his head and some experience in writing clearly for commercial clients. o_o