That's up to the DM.
As usual, there's no single clear answer to anything that isn't explicitly stated in the rules. A DM could certainly decide that waves represent an uncertain surface that the PCs will have to make rolls to move across; but they could easily rule the other way, since it's magic that says you can move across water 'as if it were solid ground', and solid ground is not generally known for heaving up and down under your feet. The latter interpretation does have some basis in our world; the original Biblical example of walking on water, which presumably inspired the spell and ring, took place in a serious storm with large waves.
What's your goal?
In general, I think the real answer comes from answering the deeper question, "What do you want to accomplish by calling for rolls?"
If there's a fight or other challenge happening and you, as the DM, want the waves to count as an environmental problem that impacts the PCs but not their aquatic enemies (thus increasing the difficulty level of the encounter), then I think that makes a pretty great fantastical setting for the scenario.
By contrast, if you're considering just having the PCs roll some checks to cross the stormy area, but those checks don't come with any actual consequences for failure (usually taking a longer time to cross an area and looking like an idiot while doing it aren't actually consequences), probably just skip it and move on to the next point of interest. You can describe them stumbling and sliding across the waves if you want to have a comedy beat, I suppose, but calling for checks in this scenario sounds a lot like the classic newbie-DM mistake of having the players make tons and tons of inconsequential rolls.
It's also worth asking yourself if your plan is eliminating the benefit of the magic item in question. If the ring is allowing the player to walk on water, but you're making them functionally perform the same rolls you'd call for from a swimming character, then you're kind of taking away the coolness and benefit of having a magic item that's perfectly suited to this challenge, and that's usually a bad thing.