Avalon1,
First, the lead officer in the case has been repeatedly quoted as saying that, in his opinion, Kennedy wasn't in the car when it went off the bridge. When he finished his "tryst" with Kopechne, she drove off alone.
Back to your question, if Chappaquiddick hadn't occurred something else most certainly would have and that something else would have aborted Kennedy's presidential chances. For various reasons, many of them understandable, Kennedy's personal life during the period was a slow motion train wreck and the press was eagerly circling.
People tend to point to Watergate as the point where the press began it's adversarial relationship with national politicians but in reality the shift from "cozy and cosseted insiders" to "crusading outsiders" had been taking place for years. Watergate was just the tipping point, the event that made the general public aware of the "new" press.
This less deferential press would have discovered Kennedy in some compromising situation sooner or later. It was just a matter of time.
Bill