LONDON — Blunt speaking made Nigel Farage, a former leader of the populist, right-wing U.K. Independence Party, one of Britain’s best-known politicians and it helped him persuade Britons to vote to quit the European Union.

But after accusing Brendan Cox, the husband of the murdered British lawmaker Jo Cox, of supporting an extremist organization, Mr. Farage faced possible legal action on Wednesday, as well as questions about his judgment.

The dispute arose on Twitter and in a radio interview in the charged aftermath of Monday’s attacks in Berlin, in which 12 people were killed and dozens more wounded. It culminated in angry exchanges and claims from Mr. Farage that an organization that campaigns in Britain to combat far-right groups, is, itself, extremist.

A critic of both the European Union and of European immigration policies, Mr. Farage responded to news of the attacks in Berlin with a Twitter post that sought to lay blame on the German chancellor, Angela Merkel, for the welcome she offered to refugees from Syria and elsewhere in 2015.