Charlie Elphicke, MP for Dover and Deal, said allowing the protests to go ahead on Saturday was "irresponsible, reckless and wrong".
Three men were arrested as groups demonstrating against immigration clashed with anti-racism protesters.
Kent Police said the law obliged it to facilitate peaceful protest.
Fights with bricks and bottles broke out as the two demonstrations met head-on, despite police attempts to keep them separate.
Right-wing marchers carried banners including the North West National Front, neo-Nazi organisation Combat 18 and the Scottish Defence League.
Counter-demonstrators included Kent Anti-Racism Network, anti-fascist group Antifa and people with Socialist Worker placards.
Six men were also arrested at a service station on the M20, where several coaches were damaged.
Eight of the men arrested have been bailed and one has been released without charge.
Police seized weapons including a knife, a knuckle duster, poles adapted to cause harm, pieces of wood, glass, hammers and bricks at the two locations.
Mr Elphicke said Kent Chief Constable Alan Pughsley had serious questions to answer about why the Dover demonstrations were allowed to go ahead.
"The police should have banned these demonstrations," he said.
Kent Police said its role was to facilitate peaceful protest and minimise disruption to the local community.
"Acts that stepped outside of this were dealt with in a proportionate manner in line with the law," it said.
"Kent Police spent a significant amount of time planning and preparing for the demonstrations, and had sufficient resources on duty to keep the public safe and to respond swiftly and effectively to spontaneous disorder."
Meanwhile, South East MEP Janice Atkinson, an independent, criticised Diane Abbott, shadow secretary of state for international development, who addressed anti-racism demonstrators before the marches began.
"Get back to Hackney Ms Abbott and take your nasty friends with you," she said. "And Nazis, you have no place in our culture, go home."
Ms Abbott's office has not yet responded.
Police have been criticised by an MP for allowing rival demonstrations to take place in Dover after they erupted into violence.