It's plausible that White Walkers (The Others in the books) can cross water (either by ship or by magically freezing the water, creating a walkway for the army of the dead).
There is this quote from A Game of Thrones (book 1):
Mormont was deaf to the edge in his voice. "The fisherfolk near Eastwatch have glimpsed white walkers on the shore."
And in A Dance with Dragons (book 5):
Cotter Pyke had made his angry mark below. “Is it grievous, my lord?” asked Clydas. “Grievous
enough.” Dead things in the wood. Dead things in the water. Six ships left, of the eleven that set sail. Jon
Snow rolled up the parchment, frowning. Night falls, he thought, and now my war begins.
That being said, just because orchestrating a naval troop movement of the army of the dead (wights) is possible for the White walkers/The Others, we can only assume that, for some reason, they can't, based on the fact that they haven't done it (yet).
The canon surrounding The Others and magic is deliberately mysterious (ATOW), so a lot of this can't be more than speculation, but the leading theory is that The Wall isn't literally just a wall of ice, it is believed to be imbued with magic that is keeping the White Walkers/The Others (but not the Wights, they seem to be immune) away, on the northern side of The Wall.
There is this quote of Bran in A Dance with Dragons
cannot pass so long as the Wall stands and the men of the Night’s Watch stay true, that’s what Old Nan
used to say. He came to meet us at the Wall, but he could not pass. He sent Sam instead, with that
Therefore we can assume (from the previous assumption that they cannot cross the water) that the magic doesn't end abruptly at both ends of The Wall, but that it has some kind of effect that, ultimately, keeps them stuck north of The Wall (whether that means the magic field extends further out into the sea, and how that's detailed, is even less clear).