Mr Biden, asked by reporters in Delaware why Mr Obama had not endorsed him, replied: "I asked President Obama not to endorse and he doesn't want to… whoever wins this nomination should win it on their own merits."
He is the most experienced of the Democratic candidates. A six-term senator, he ran twice unsuccessfully for president - in 1988 and 2008.
Mr Biden was tipped to run for president again in 2016, the year Mr Trump was elected, but ruled himself out after the death of his 46-year-old son, Beau Biden, from a brain tumor.
He has enjoyed relative popularity among Democrats in recent years, consistently leading every national poll of the Democratic primary tracked by the website RealClearPolitics.
But the former vice-president has been forced to address claims he inappropriately invaded the personal space of women.
He apologized, saying: "The boundaries of protecting personal space have been reset. I understand it and I'll be much more mindful."
Joe Biden enters the Democratic presidential contest as a front-runner, with huge name recognition and the potential to raise vast amounts of campaign donations.
Of course, so did Hillary Clinton in 2015 - and we all know how that turned out.
The former vice-president has a lot going for him. He also has a lot going against him. The durability of his campaign is one of the big questions hovering over the Democratic race.