It would be a Strength (Athletics) check to swim across.
There are two parts to this.
First, lets throw Dexterity away. Just looking at the situation before the rules, what does Dexterity have to do with resisting the current of the river? Even maintaining balance when attempting to wade through is a stretch. The only argument is that the force is from the side and the movement isn't directly against the current as it would be, say, walking into Gust of Wind. Looking at the Player's Handbook, pg 175 on Ability Scores, under Athletics:
You struggle to swim or stay afloat in treacherous currents, storm-tossed waves, or areas of thick seaweed. Or another creature tries to push or pull you underwater or otherwise interfere with your swimming.
Now, variant ability scores, of which I am a fan, would allow other abilities to use Athletics proficiency, but again, Strength makes more sense. The section on strength says it should be used when trying to "force your body through a space." Dexterity mentions "tricky footing" but this is resisting the force of the flow, not trying to walk on slippery, wet rock.
From the same chapter on pg 179, Saving Throws are to resist something done to the character, as opposed to Ability Checks, which are to resolve actions taken by the character.
A saving throw, also called a save, represents an attempt to resist a spell, a >trap, a poison, a disease, or a similar threat. You don't normally decide to >make a saving throw; you are forced to make one because your character or >monster is at risk of harm.
So while a Strength save might be appropriate to initially resist being swept away, a success leaves the character where they fell into the water. It would take a successful Strength (Athletics) check to swim across, and failure should probably result in being pushed downstream.