No.
The prize is granted by the Nobel Peace Prize Committee. For the Peace Prize - there's no involvement of the Norwegian King at all. While the Swedish presentation ceremonies include the Swedish King presenting the prize, the Nobel Peace Prize is presented in Norway, by the committee chair. The King's role is purely ceremonial, in that that he's present at the ceremony.
According to the US DOJ:
The Foreign Gifts and Decorations Act does not bar the President from accepting the Nobel Peace Prize without congressional consent, because the Norwegian Nobel Committee is not a “unit of a foreign governmental authority,” an “international or multinational organization whose membership is composed of any unit of foreign government,” or an “agent or representative of any such unit or such organization.”
The full opinion has references to authorities.
Some observations from the memo:
since 1906, there have been at least six federal officers who have accepted the Prize while serving in their elected or appointed offices. The Peace Prize has been received by two
other sitting Presidents—Theodore Roosevelt and Woodrow Wilson—by a sitting Vice President, Secretary of State, and Senator, and by a retired General of the Army, with the most recent of these acceptances having occurred in 1973. Throughout this history, we have found no indication that either the Executive or the Legislative Branch thought congressional approval was necessary.
...
President Wilson ... deposited the award money in a personal account in a Swedish bank. ... President Wilson does not appear to have sought congressional approval for his acceptance, nor does it appear that Congress thought its consent was required.
