Cardinal Walter Kasper, 77, made the remarks in a German magazine interview.
The Vatican said the cardinal had not intended "any kind of slight", and was referring to the UK's multicultural society.
It added that he had simply pulled out of the Pope's visit due to illness.
The German-born cardinal was quoted as saying to the country's Focus magazine that "when you land at Heathrow you think at times you have landed in a Third World country".
He also criticised British Airways (BA), saying that when you wear a cross on the airline "you are discriminated against".
In 2006 a BA employee was told to stop wearing a cross at work. She took the case to an employment tribunal claiming religious discrimination, but lost, also losing her subsequent appeal.
BA changed its uniform rules in 2007, allowing staff to display a faith or charity symbol.
Vatican sources said Cardinal Kasper - who stepped down in July as the head of the department that deals with other Christian denominations - was suffering from gout and had been advised by his doctors not to travel to the UK.
Not all of the cardinal's comments in the interview were critical of the UK.
He also said: "Everyone who knows England knows that there is also a great Christian tradition there."
The Pope is spending four days in Scotland and England, starting on Thursday.
The BBC's correspondent in Rome, David Willey, said the cardinal's reported comments were "a slightly clumsy thing to have done on the eve of the visit".
However, he added that he did not think it would have much effect on the Pope's trip to the UK.
Clifford Langley, from Catholic newspaper The Tablet, said the cardinal was "obviously talking nonsense".
"I don't think he believes Britain is in the grip of secular atheism, and he shouldn't have said so," said Mr Langley.
"They are saying it is ill health [that has forced the cardinal to drop out of the visit], but I wonder if that is the fact. I wonder if he has been dropped because he is an embarrassment."
British Airways said the cardinal had been "seriously misinformed" in his claims about the airline.
"It is completely untrue that we discriminate against Christians or members of any faith," it said in a statement.