LOUISVILLE, Ky.—Democrats Wednesday launched the second phase of their fight to capture a long-held Republican House seat in Georgia, but the party’s attempts to unify remain hindered by lingering internal divisions.

First-time candidate Jon Ossoff, who had raised more than $8 million in a matter of months from Democrats across the country, garnered 48.1% of the vote in a crowded open primary Tuesday, just shy of the 50% threshold needed to capture the seat outright.

...