Owens, who is Welsh and gay, officiated over England's 24-21 defeat to New Zealand at Twickenham on Saturday.
A member of the crowd wrote to The Guardian newspaper to express his disgust at the "nasty, foul-mouthed, racist, homophobic abuse" and the RFU has consequently announced their intention to look into the situation.
"The RFU condemns all forms of discrimination and aims to ensure that all people, irrespective of their age, gender, ability, race, religion, ethnic origin, creed, colour, nationality, social status or sexual orientation, have a genuine and equal opportunity to enjoy rugby union in whatever form, on or off the pitch, at all levels and in all roles," a statement read.
"We take any allegations of any abuse very seriously and are investigating this matter."
Owens himself, meanwhile, believes anyone found guilty of such an offence should be heavily punished by the relevant authorities.
"If someone has gone to the trouble to send a letter to the paper and say that this was disgusting and the behaviour was totally out of order, then I would think what they shouted was pretty bad and if that is the case then these people should be banned from the game," he told The Telegraph.
"They need to be told that some behaviour is not acceptable, that crossing the line of what is humour and banter is a fine line but if you cross it then you have to put up with the consequences."