PARIS—French President Emmanuel Macron suffered the first major setback in his push to overhaul the French economy, backing off a fuel-tax increase that enraged much of the nation and sparked a grass-roots protest movement against his government.

Faced with another weekend of destructive protests by the gilets jaunes—or yellow vests—Prime Minister Édouard Philippe told a press conference on Tuesday that the tax increase would be pushed back six months to allow for public discussion. The worst riots to hit Paris in decades...