Welsh Culture Minister Alun Pugh has asked the Home Office to review the law on the display and wearing of Nazi regalia.
Prince Harry has been heavily criticised for his actions
It follows the row surrounding Prince Harry's choice of a Nazi uniform to wear to a fancy dress party, for which he later apologised.
Mr Pugh's letter came as politicians in Germany called for a Europe-wide ban on Nazi symbols.
The Clwyd West AM called people who chose to wear such regalia "dimwits".
Politicians and community leaders worldwide expressed anger and concern after the 20-year-old prince was pictured in the media at a friend's birthday party wearing a Nazi uniform.
He later issued a statement saying: "I am very sorry if I caused any offence or embarrassment to anyone. It was a poor choice of costume and I apologise."
Mr Pugh said: "Wales lost many fine men and women in the war against the Nazi tyranny.
"Just before Christmas I had a blazing row in Cardiff with a street trader who was selling Nazi flags alongside our red dragon.
"I know that many of my Clwyd West constituents will have been deeply offended to see the son of the Prince of Wales sporting the regalia of Auschwitz."
He added the only "legitimate place" for displaying Nazi memorabilia and symbols was in museums and history textbooks "in order to document the atrocities of the Third Reich and their attempts to enslave us".
"The people who sell this stuff and the dimwits who wear it would face criminal sanctions in other parts of Europe and I think we should examine the case for doing the same."
Wolfgang Bosback, vice-president of Germany's Christian Democratic parliamentary group, said it was possible European justice ministers would discuss bringing in a European prohibition on displaying the swastika and other Nazi signs.
The symbols are banned under German law.
The Liberal group in the European Parliament has said all of Europe suffered because of the crimes of the Nazis, so there should be a continent-wide ban.