(Adds Sarkozy)

PARIS, June 26 (Reuters) - The European Union must stop being nit-picky and intrusive, French Prime Minister Manuel Valls said on Sunday as the bloc scrambled to handle the aftermath of Britain’s vote to leave.

The Brexit vote to leave the EU has deepened fears among mainstream politicians across Europe about the rise of eurosceptic, anti-establishment parties, particularly in France, where the far-right National Front is increasingly popular.

That concern has also prompted mainstream French politicians on the left and right to call for an overhaul of Europe, in a country where surveys show disillusionment with Brussels is growing.

“We must put an end to this sad and finicky Europe. Too often it is intrusive on details and desperately absent on what’s essential,” Valls said. “We must break away from the dogma of ever more Europe. Europe must act not by principle but when it is useful and pertinent.”

Former President Nicolas Sarkozy echoed those sentiments. “Europe cannot continue to build itself against the people,” he said.

While a large majority of French are attached to EU membership, polls over the past few years have consistently shown a majority also want Europe to have fewer powers, said Francois Miquet-Marty, head of the Viavoice polling institute.

Only one on five voters want “more Europe”, he said.

Long in the EU’s driving seat alongside the bloc’s power-house Germany, France, dogged by a poor track-record on abiding by the EU’s fiscal rules, has been steadily less active on the EU scene over the past years.

In his speech on Sunday in Normandy, Valls insisted Europe “is not about seeing sovereign states disappear.”

He also said Europe must defend its economic interests better, on trade matters but also on financial, social, environmental and cultural issues. “Europe must be firm,” he said.

While an Odoxa opinion poll on Friday showed 64 percent of French want to stay in the EU, the number of those who want to leave rose by 7 percentage points in the course of three months.

Sarkozy said he wasn’t sure what the answer would be if the French were asked a Leave or Remain question in a referendum.

“Do you think that 30 kilometres of Channel between us mean we’d think that differently?” he told France 2 television.

However Sarkozy, who had so far rejected the idea of an EU referendum in France, added his voice to a growing chorus of politicians of all political stripes saying a referendum on a new treaty or EU project could be a good idea.

Only the far-right National Front wants a referendum on EU membership.

“We should not be afraid of the people,” Sarkozy said. If there was a new treaty setting out clear choices for Europe “I don’t see how we could refuse (a referendum),” he said. (Reporting by Ingrid Melander; Editing by Stephen Powell and Alexandra Hudson)