Amazon founder Jeff Bezos had contested the decision, in part because he said the contract was supposed to have been awarded to more than one bidder. However, a funding shortfall from Congress meant this wasn't possible, according to a rationale published by Nasa at the time of the contract announcement.
Mr Nelson partially blamed the landing mission's delay on the lawsuit.
"Returning to the Moon as quickly and safely as possible is an agency priority. However, with the recent lawsuit and other factors, the first human landing under Artemis is likely no earlier than 2025," he said.
However, commentators had been saying since last year that the lander cash problem alone made the 2024 date untenable.
Mr Bezos' firm Blue Origin had partnered with three other aerospace companies to vie for the prestigious lander contract.
The judgment last week means that a version of SpaceX's Starship - currently undergoing testing at a site in southern Texas - will be the vehicle used to carry people down to the lunar surface on that mission.
The first mission under the Artemis programme is set to fly in February next year. Nasa will launch the Orion spacecraft on the powerful Space Launch System (SLS) rocket without people aboard.
During this mission, Orion will fly around the Moon on a voyage lasting three weeks in order to test its systems.
The first flight with astronauts - Artemis-2 - will now follow in 2024, Mr Nelson said. It will also fly around the Moon.
Artemis-3 will be the first mission to return to the surface of the Moon since Apollo 17 in 1972. It is set to land at the lunar south pole, which is thought to hold vast stores of water-ice in craters that never see sunlight.
The ice in these craters could be used to make rocket fuel on the Moon, bringing down the cost of lunar exploration because it would not need to be shipped from Earth.
The programme will also see the first person of colour land on the Moon, though it is unclear whether this will happen during Artemis-3 or a later mission.