A daily newspaper which published the Pontypridd MP's comments has also quoted him as saying that the royal family should be self-sufficient, and should not be subsidised by the taxpayer.
Dr Howells is said to have told The Daily Telegraph that he had "never understood the attraction of royalty".
And he added: "This isn't the first generation. They're all a bit bonkers.
"They choose very strange partners, they're not managing the modern world very well."
He has so far refused to make any further comment.
At his home in Pontypridd, south Wales, he refused to be interviewed, telling reporters:"I've said all I'm going to say. I've got no further comment."
He also reacted to allegations that Prince Edward, the Earl of Wessex, had used official overseastrips to secure business for his film company - allegations strongly denied by the Palace.
Mr Howells said: "I suppose everybody trades on their fame, but I don't think the taxpayer should be paying for royal visits and first class air tickets to increase their profit margin."
And he had something to say about the Earl's brother, Prince Charles. "We don't see him in Wales very much," he said.
The minister's comments have provided additional fuel for the fire of controversy surrounding the unguarded opinions expressed by Prince Edward's wife Sophie Rhys-Jones during a business meeting with a reporter posing as a sheik.
Fellow Welsh Labour MP Paul Flynn - who has long campaigned for an elected head of state - rushed to defend the Countess against media criticism.
The MP for Newport West said he deplored the actions of "the jackals of the tabloid press".
But, he added:"Life for a modern royal is like living in a monastery with glass walls.
"They are under merciless surveillance by the media. We expect full-blooded human beings to live saintly lives.
"It's as bad as being condemned to spend a life sentence in the Big Brother house.
"The real world and the make-believe royal world are irreconcilable. They cannot stay aloof from politics or business and simultaneously be political and entrepreneurial.
"It's a cruelty to condemn another generation to spend their lives from cradle to grave in the goldfish bowl, exposed to the merciless mockery of the media."
On Sunday The News of the World printed six pages of secretly taped conversations between Sophie, her business partner Murray Harkin and undercover reporters, in which she allegedly made disparaging comments about leading royals and politicians.
Worst week
According to the paper, the Countess said Chancellor Gordon Brown's Budget was "a load of pap", Cherie Blair "hates the countryside" and Conservative leader William Hague "sounds like a puppet".
The publication adds to what has been one of the worst weeks for the Royal Family since the controversy that followed the death of Diana, Princess of Wales.
Other newspapers have followed up the indiscretions, particularly those allegedly connecting Mr Harkin to drug-taking.
In the conversation with NoW's reporters - at which Sophie was not present - Mr Harkin allegedly said "I don't do lots of drugs but the odd line of coke I quite like" and admitted taking ecstasy some years ago.
The paper consulted its lawyers and decided to go back on an earlier agreement with Sophie not to publish the secretly taped conversation with the Countess.
NoW managing editor Stuart Kuttner defended the decision to publish, insisting on BBC1's Breakfast with Frost programme that the newspaper had acted to "partly vindicate, partly exonerate" Sophie after a week of inaccurate reports.
"For six or seven days all the media recycled and churned out false and inaccurate quotes - we took a view it was time to clear the air, get the words down, get it right and in a sense partly vindicate, partly exonerate Sophie."