The authorities have issued a decree closing 131 media outlets while scores of top journalists have been arrested.
The Lib Dems said the "purge" of critics of the government was contrary to the principles of democracy, liberty and the rule of law upheld by Nato.
Turkey is a key member of the defence alliance and ally of the United States.
The Turkish government has been accused of using the coup attempt by sections of the military as a pretext to target journalists and other non-state institutions including judges, academics and teachers.
Three news agencies, 16 TV channels, 23 radio stations, 45 papers, 15 magazines and 29 publishers are to be closed down.
Many of the media outlets are linked to the Hizmet movement of US-based cleric Fetullah Gulen, whom Turkey's democratically elected president Recep Tayyip Erdogan has blamed for the uprising, a claim Mr Gulen denies.
Despite strongly condemning the coup and giving its backing to President Erdogan, the international community has told the government its response must be proportionate, follow due process and avoid indiscriminate reprisals.
But the Lib Dems, who served in coalition government with the Conservatives between 2010 and 2015 but were reduced to eight MPs at the last election, have urged Nato to act.
Tom Brake, the party's foreign affairs spokesman, said the flurry of arrests and closure of media outlets in recent days "should send shivers down the spine of any person who believes in a free and open society".
"Erdogan's ongoing purge of newspapers, academics, teachers and judges has nothing to do with Turkey's security and everything to do with blocking any opposition to his increasingly authoritarian rule," he said in a statement to the Lib Dem voice blog.
"The preamble to Nato's founding treaty refers to it being "founded on the principles of democracy, individual liberty and the rule of law", all of which are under threat in Turkey currently.
"If the UK and our Nato allies want to protect these core principles, it is time to make it clear to Erdogan that his actions will have lasting international consequences, and I am calling on Nato to urgently consider suspension of Turkey's membership."
He told the BBC that the UK - a founding member of Nato - should take a firm stance on the issue and should be insisting that those who had been detained had access to legal representation and anyone who was not charged was speedily released.
"I hope the new government will use this as an opportunity to demonstrate that human rights are at the core of our foreign policy and will not be downgraded in favour of international trade," he said.
Turkey already has a poor track record on media freedom, ranking 151 out of 180 countries in this year's World Press Freedom Index.
At least 246 people were killed and more than 2,000 injured in clashes during the abortive coup - 16,000 people have been detained since then.
Turkey has declared a three-month state of emergency, allowing the president and the government to bypass parliament when drafting new laws and to restrict or suspend rights and freedoms.
Earlier this year the EU called for longstanding talks on Turkey becoming a member to be re-energised although observers have said the coup and the government's response have set back the prospect of any immediate progress.
EU officials have said the accession talks will be suspended immediately, if the Turkish government reintroduces the death penalty.
Nato should consider suspending Turkey, the Lib Dems have said, following its crackdown on the media and other bodies in response to the recent coup attempt.