Complying to the yes or no condition imposed, the answer is NO. There is no rule, or law blocking Bernie Sanders to run for president independently. Bernie Sanders can run for presidency as an independent.
Sure, he would not be able to get any votes in states with sore loser laws, or in states where the deadline has passed,
which would guarantee he would lose even if he wins 100% of the votes in the states that would allow him to run, and 100% in the states that allow write-ins, but he has the option.
As long as there are states that allow write-ins, there will always be this option for literally every natural-born US citizen.
Edit: Please see the comment for a way in which Bernie Sanders could win, even if he does not even run in most states, but only in those that would still allow him to run as an independent.