PHILADELPHIA — Joseph R. Biden Jr., sometimes known to Democrats in this city as Pennsylvania’s “third senator,” returned to the state of his birth on Saturday to deliver a forceful call for national unity, looking past the Democratic presidential primary to directly appeal to the voters who helped power President Trump’s victory in Pennsylvania and across the country in 2016.

Mr. Biden, a former vice president and former Delaware senator who is now pursuing his third bid for the presidency, trained his eye squarely on the general election. He cast the contest against Mr. Trump as one for the soul of the country, combining calls for more civility in American politics with bitter denunciations of Mr. Trump and the impact he is having on the nation.

And Mr. Biden struck a defiant tone toward those in his own party who have expressed discomfort with his emphasis on bipartisanship and his legacy of Washington deal-making. He argued that the stakes of the coming presidential election should transcend partisan passions of the moment.

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“They say Democrats are so angry, the angrier a candidate can be, the better chance he or she has to win the Democratic nomination,” Mr. Biden said. “Well, I don’t believe it, I really don’t.”