WASHINGTON—Former Vice President Joe Biden opened his presidential campaign Thursday by pointing to a “battle for the soul of this nation,” in what may be the last major addition to a sprawling lineup of Democratic candidates competing to challenge President Trump in 2020.

Mr. Biden’s entry means the Democratic contest is largely set, putting the former vice president into a scramble for the nomination against a diverse field of candidates who have been competing and raising money for months. The contest already is revolving...