Will the US and Russia start World War III?
This sounds more like a joke than a question

Last war United States won, or was part of the winning side, has been World War II that ended in 1945 (77 years ago). Ever since then US has not faced any actual military force and any other smaller conflict with armed groups US faced they lost miserably. The latest being the war against Taliban that ended with a great escape and a donation of $100 billion worth of weapons and military equipment to Taliban to buy safe passage out of Afghanistan.
Going to war with an actual adversary will not only destroy US military but also US and EU economy. That's why it is never going to happen.