"Mr. Christopher remains and will remain a US citizen," a Royal Court statement said on Friday.
O'Neill, a New York-based financier who holds both US and British citizenship, will also continue working, which clashes with the expectations of members of the Swedish royal family who work only for charitable causes and representing the state.
Refraining from business and becoming a Swedish national are "principles" of membership of the royal family, the statement read.
"In reference to these requirements, Mr. Christopher O'Neill has respectfully asked to remain a private citizen and not be granted royal status," Marshal of the Realm Svante Lindqvist wrote.
The request means he will not be entitled prince, and neither will he become Duke of Gästrikland and Hälsingland.
The couple's official monogramme was also revealed on Tuesday, with M for Madeleine supported by two Cs on either side. Designed by heraldry artist Vladimir A. Sagerlund, Madeleine's duchess crown "rests atop her initial M", the statement read.