"Eventual" is probably not the best term to use as it implies the continent will stop moving at some time in the future. Right now, and for the next 100 million years or so, North America is moving farther from Eurasia. The growing separation is pretty much along an east west axis. Further into the future than that is vague because of the many variables like the continued motion of the continents and the development of new convective currents or their inactivity.
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There is no definitive answer, as we can only theorize about distant future. We can calculate with a certainty that the Atlantic will continue to widen in next several millions of years or even tens of millions. However, we are not sure, what are other forced in motion all over the world’s tectonic, and what new forces may arise from within deep mantle. So far, there been proposing two theories, which one estimating North America crashing with Asia, while the second claims that the Atlantic will close as a new subduction zone will emerge along the coast of the Northern America.
The first theory has several issues. First Americas are moving west, but also they bit rotate, with Greenland moving toward north-northwest. At the same time western coast of the continent is continued toward Alaska, as eventually seems California heads that way. However, there is no general agreement, what will happen with the eastern Siberia, that is part of the Northern American plate, and other plate in the region (Amurian) is being debated. This location been geologically stable in past 50 million year; therefore, it is difficult to estimate, if the forces can be put again into motion in 50 or even 100 million years from now.
Second theory pushes the Northern America over the remains of the Farallon plate, and the subduction zone from the past lies buried deep beneath continent. Some propose that the Madrid fault is the one, as the continent ride over it and eventually will continue eastward and appear in Atlantic. Question remains, if the subduction zone will open again and will start swallowing Atlantic seafloor, and therefore, consuming the wast ocean and close the gap again and crashing the continent back to Africa and Eurasia. As with the first theory, the there is issue of the Caribbean plate that is thicker and have not been sudbucted, in the past, and southern American and Cocos plate subdue beneath it. If the Atlantic will get smaller, what would happen with it?
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