Weber's demand came on the eve of key EU/Turkey talks in Brussels on Tuesday and amid a mounting feud between Berlin and Ankara.
With the EU/Turkey high level political dialogue due to take place on Tuesday, Weber said Turkey's EU accession "makes no sense."
He stated, "The EU should set itself the goal of ending the accession talks."
The talks, led by EU foreign affairs chief Federica Mogherini and Turkish foreign affairs minister, threaten to be overshadowed by protests in Brussels about alleged violations of human rights in Turkey.
The demonstrations will include Özlem Dalkıran, who works for Avaaz, an internet-based civic organisation. He is one of the human rights defenders who were arrested recently in Turkey.
Amnesty International is also due to demonstrate in the EU quarter of Brussels on Tuesday.
Its Secretary General, Salil Shetty, will urge Mogherini to urgently raise the issue of jailed human rights defenders - including Amnesty International's Turkey Director and Chair, Idil Eser and Taner Kılıç - at her meeting with the Turkish side.
Shetty will then join scores of activists at a protest.
"Governments across the world have called for the release of human rights defenders in Turkey. I will press upon Federica Mogherini that she must do the same, and urgently raise the issue with Minister Çavuşoğlu when they meet in Brussels on Tuesday," said Shetty.
"Defending human rights is not a crime, but it is fast becoming one in Turkey. We only have a small window of opportunity to stop this now, before there's no one left to stand up for what's right."
The EU/Turkey meeting comes against a backdrop of Turkey's current spat with Germany.
Berlin has lambasted the government of Turkish President Recep Tayyip Erdoğan, and German Chancellor Angela Merkel's chief-of-staff, Peter Altmaier, called Ankara's behaviour "unacceptable" after Turkish police arrested a German in a swoop on human rights activists.
EPP group leader Manfred Weber, a German MEP, said a close and privileged relationship between the EU and Turkey is in the interest of both sides and is of strategic importance for coping with the challenges both sides face.
"However," he added, "the EPP group firmly believes that the privileged special relationship between EU and Turkey should be based on the rule of law and on the respect of the fundamentals freedoms and democratic values."
Germany's finance minister Wolfgang Schäuble, speaking of the current dispute, said President Erdoğan "is jeopardising the centuries-old partnership" between Germany and Turkey, adding, "We can't allow ourselves to be blackmailed."
European Economic and Social Committee President Georges Dassis has also voiced his concerns about the current situation in Turkey regarding the respect for fundamental rights.
"The EESC expresses its solidarity with the Turkish people and gives its full support for the organisations of Turkish civil society which defend democracy.
"We hope we will once again see a Turkey that is totally democratic, open to the future and respectful of all freedoms, the rule of law and the separation of powers," he said.