He jumped off a cliff into the water, there was no shore on sight, at least not a shore that wasn't cliffsides, and he even had to swim away from a Gyarados that made its home in that river. Exactly how would he have gotten out anyways? Not to mention he has to keep hidden from the flock of Spearow he pretty much infuriated earlier (heck, that was the reason he even jumped in earlier).
Even if he DID somehow manage to get out of that body of water without Misty's help, pretty much the entire first episode nonetheless showed that he is completely incapable of travelling by himself, meaning he definitely wouldn't know how to cook to save his own life, meaning he definitely would be dead meat, literally, by the time he got to Viridian Forest via starvation (as it apparently took them 13 days just to get from Viridian City to Pewter City, based on what Showdown on Pewter City implied).
And BTW, there were plenty of instances where if it weren't for Misty, Ash at the very least would NOT have been able to get a necessary badge for the league (Riddle Me This is a good example, considering Misty's pretty much the reason why Ash even found Cinnabar Gym, never mind won the Volcano Badge by beating Blaine, and I should remind you that Cinnabar Gym was believed to have been shut down), and at worst would have died (aside from the events of the first episode, there's also what happened with the St. Anne when she, Ash, Brock, Jessie and James got trapped in there. She actually managed to come up with the plan to get them out of there, or at least her group). Heck, she even actually stood up and did something in episodes rather than simply complaining, and during the Pewter Gym match, she even clearly attempted to help Ash by supplying him with her Water Pokémon (only for Ash to refuse). She doesn't spend her time complaining the entire journey.
And anyways, to everyone else in the topic, hasn't anyone seen what happened when he tried to journey by himself in the first episode? He nearly ended up dead. So it's clear what would have happened if he tried to journey by himself.