It depends when and where this accident happens, and what kind of accident we are talking about. If a US nuke goes off on its own territory (e.g. a Goldsboro 1961 scenario), does it lead to nuclear war? Well, probably not, if it's clear what happened and things are smooth enough to sort out the facts of it.
What if it happens during the Cuban Missile Crisis, or during the Able Archer 83 crisis? Well, then you have a pretty bad situation, because escalation potentially kicks in very quickly.
The time matters a lot because both US and Soviet abilities to sort out sense from nonsense varies, and because the varying states of the early warning systems and the arsenals means that both states felt like they were in a "use it or lose it" situation with regards to many of their nukes (if an attack is incoming, they felt that if they didn't immediately fire a retaliation, then they lose the nukes in question).
The "decision-making" situation also changes over time. At times the US system is slow and requires lots of deliberate orders for things to happen. At times, it doesn't. (Ditto the Soviets.) The times in which it doesn't require a lot of deliberate orders are the most dangerous from the position of something accidental. (The US never made their system fully automatic and computerized, because even the most gung-ho generals, even the ones most worried about losing their ability to retaliate, realized that the computers were too fallible to trust 100% on this point.)
The period in question also affects the communication methods. Let's say a nuke accidentally goes off. In the early 1960s or late 1950s, we literally have no quick and easy way to tell the Soviets what is going on rapidly. The entire communication system used during the Cuban Missile Crisis are civilian telegram networks. Even the so-called "hot line" that gets installed afterwards is still just piggybacking on nuclear-vulnerable civilian telecommunication lines. If an "averted nuclear weapons use" scenario requires perfect or even good, rapid communications between the superpowers, this is not a guarantee at many points in history.
In the 1963 Secretary of Defense McNamara seriously fretted about this:
Mr. McNamara went on to describe the possibilities which existed for an accidental launch of a missile against the USSR. He pointed out that we were spending millions of dollars to reduce this problem to a minimum, but that we could not assure ourselves completely against such a contingency. Moreover he suggested that it was unlikely that the Soviets were spending as much as we were in attempting to narrow the limits of possible accidental launch. He went on to describe crashes of US aircraft[,] one in North Carolina and one in Texas, where, by the slightest margin of chance, literally the failure of two wires to cross, a nuclear explosion was averted.
McNamara was no anti-nuclear wimp — if he's seriously worried about accidental nuclear war at this time, it's probably a real problem.
The most dangerous period was the late 1950s, early 1960s, where the weapons systems are big enough and sophisticated enough for serious amounts of damage to result if things go wrong, but not sophisticated enough to guarantee against error or bad communication. The late 1970s/early 1980s are also dangerous because of the tensions between the powers, the massive and fast nature of the arsenals, and the numerous errors that get introduced by buggy computerized systems. What makes the Petrov/Arkhipov scenarios most frightening is that they occurred during crisis periods, which are the most vulnerable to major misunderstandings and fears.
Schlosser's Command and Control is a must-read for making full sense of how these systems have evolved over time.
The problem with any answers to a question like this is that we simply don't know. It's a hypothetical whose mystery was a major cause of 50 years of conflict between the US and Soviet Union. I'm sure you can find sources of this or that leader saying one thing or another, but when the cards are on the table and someone is faced with the annihilation of millions of people, there's just no way to know. However, if you want a little bit of insight into the thinking of the time, The Cuban Missile Crisis and The Dead Hand can give you an idea to the more extreme side of eventualities
Sorry, but your submission has been removed because we don't allow hypothetical questions. If possible, please feel free to rephrase the question so that it does not call for such speculation, and resubmit Otherwise, this sort of thing is better suited for /r/historicalwhatif.