The Stone MP, whose Bridgnorth-based son William is a UKIP supporter, was jeered by some Labour MPs when he made the comments during questions on foreign national offenders in the House of Commons.
Mr Cash, whose father, Captain Paul Cash, was killed in Normandy on July 13, 1944, said: "Those who fought and died in the Normandy landings did not do so to enable convicted EU rapists, paedophiles and drug dealers, who are now here in prison, to be protected under the new European human rights laws."
He added that such offenders "should be deported" and said the Home Affairs Select Committee "was clearly right" to suggest the public would question the point of the UK remaining in the EU.
Mr Cash's comments drew criticism from Labour's Shadow Home Secretary Andy Burnham, who accused him of making an "opportunistic election broadcast on behalf of the leave campaign". He added: "This is not an EU question, but a question of the competence of the Government."
And former Chancellor Kenneth Clarke, said: "If we were not members of the EU, we would go back to a system where we couldn't deport anybody to the country of their origin unless we could persuade the government of that country to accept them." But Home Secretary Theresa May said she recognised that Mr Cash had "his own personal reasons for remembering the D-Day landings".
She added: "And it is true that those who gave their lives on the beaches of Normandy did so to protect our freedoms. This Government has put in place a number of measures and we continue to work to do more to ensure we can protect the public from these serious criminals, rapists and others, who may choose to come here from whichever country in the world."
Mr Cash also received a wrap over the knuckles from the House of Commons speaker for going "off script" during an urgent question. Speaker John Bercow told Mr Cash to "stick to the terms of the question".