Two decades after Tony Blair swept to power in a landslide victory to become prime minister of Britain, his brand of centrist, pro-European politics seems a distant memory in a divided country that has now voted to leave the European Union.

But as the anniversary of that victory approached, Mr. Blair tried to bolster the sagging center ground of British politics, while predicting that voters might think again about the economically damaging hard break with the European Union that he believes is looming.

Britons will vote on June 8 in a general election called by Prime Minister Theresa May, a Conservative, who wants a larger parliamentary majority behind her to negotiate the withdrawal from the European Union required by a referendum last year.

Mr. Blair, in an hourlong conversation with the international news media, urged Europe’s left not to reject globalization, while warning that Mrs. May would probably lead Britain to a sharp break with the European Union, or a “hard Brexit.” Yet he also argued that once the shape of any departure deal becomes clear, Britons may have second thoughts.