In the case of a Presidential nominee, I would think the second strongest primary candidate would demand the nomination (particularly when that candidate is Bernie).
Your Eagleton post is a good reminder of how much things have changed.
All Eagleton did was see a shrink for depression—he didn’t steal anything, run off with thousands of classified documents, run an international extortion slush fund, kill political enemies, none of it.
I’m certain that the rules require majority vote. So no, the 2nd place finisher does not get an automatic nod.
Getting a hold of Democrat Party rules is not easy. However I would be shocked if, in the event of the VP candidate’s resignation, that the POTUS candidate could alone nominate a replacement.
Other relevant case: Death of Horace Greeley....most of his electors (but not all) voted for his VP candidate, B. Gratz Brown, as president.
Kaine would have a greater claim to the nomination than Sanders.