In essence, the system of mandates was a compromise between the Allies’ desire to retain the former colonies of Germany and Turkey, and their prior acceptance of President Woodrow Wilson's Fourteen Points, which effectively declared that annexing territory had not been the aim of the allies in the war.
The allies had issued their Conditional Acceptance of the Fourteen Points on 5 November 1918.
It is also worth noting that, had the former German and Ottoman territories been ceded to the victorious powers directly, their economic value might have been credited to offset the Allies' subsequent claims for war reparations.
The Mandate system was created by Article 22 of the Covenant of the League of Nations.
Tellingly, the League of Nations Official Journal for June 1922 includes a statement by the British Prime Minister, Lord Balfour, in which he made the point that the League's authority was strictly limited in the Mandates:
Mandates were not the creation of the League, and they could not in substance be altered by the League. The League's duties were confined to seeing that the specific and detailed terms of the mandates were in accordance with the decisions taken by the Allied and Associated Powers, and that in carrying out these mandates the Mandatory Powers should be under the supervision -- not under the control -- of the League.
Since the United States never ratified the Treaty of Versailles, and never joined the League of Nations, there wasn't a great deal that they could do to challenge the implementation of the system.