The parents of Madeleine McCann are “naturally disappointed” at losing their legal battle over comments made by a retired Portuguese detective but said they only took action to protect the search for their daughter.

Lawyers for Kate and Gerry McCann had argued that Portuguese authorities had breached their right to respect for a private and family life in the way the courts there dealt with their libel claims against Goncalo Amaral.

He claimed in a book, TV documentary and newspaper interview that the McCanns were involved in their daughter Madeleine’s disappearance.

In a judgment published on Tuesday, the European Court of Human Rights rejected the couple’s arguments, finding that there was extensive publicity around the claims against them before Mr Amaral published his book in July 2008.

The McCanns said in a statement issued through their lawyers: “We are naturally disappointed with decision of the European Court of Human Rights announced today.

“However, much has changed since we started legal proceedings 13 years ago against Mr Amaral, his publisher and broadcaster.

“We took action for one and only one reason: Mr Amaral’s unfounded claims were having a detrimental impact on the search for Madeleine.

“If the public believed that we were involved in her disappearance, then people would not be alert for possible clues and may not report relevant information to the relevant law enforcement agencies.

“The focus is now rightly on the search for Madeleine and her abductor(s).

“We are grateful for the ongoing work by the British, German and Portuguese police.

“We hope that, with the help of the public, hard work and diligence we can eventually find those responsible for Madeleine’s disappearance and bring them to justice.”

Three-year-old Madeleine vanished during a family holiday to Praia da Luz, Portugal, on May 3 2007.

Earlier this year German investigators said they had new evidence against prime suspect in the case and convicted sex offender Christian Brueckner, and in April he was officially declared a suspect by Portuguese officials as a 15-year legal deadline approached.

The ECHR found that Mr Amaral’s claims had not had a negative effect on the McCann’s “social relations” or their campaign to find their daughter.

It said: “While the Court understood that the book’s publication had undeniably caused anger, anguish and distress to the applicants, it did not appear that the book, or the broadcasting of the documentary, had had a serious impact on the applicants’ social relations or on their legitimate and ongoing attempts to find their daughter.”

The court also said that the McCanns’ right to respect for their private and family life had to be balanced against Mr Amaral’s freedom of expression.

The ECHR said: “The Court considered that, even assuming that the applicants’ reputation had been damaged, this was not on account of the argument put forward by the book’s author but rather as a result of the suspicions expressed against them, which had led to their being placed under investigation in the course of the criminal investigation (the prosecutor’s office decided to take no further action in July 2008) and had led to intense media attention and much controversy.

“The information had thus been brought to the public’s attention in some detail even before the investigation file was made available to the media and the book in question was published.

“It followed that the national authorities had not failed in their positive obligation to protect the applicants’ right to respect for their private life.”

The applicants’ complaint concerning their “right to be presumed innocent” was also rejected, because Mr Amaral’s book was published three days after the decision to take no action against them.

The McCanns will not appeal against the decision.