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In a 29-page ruling, they said: "We conclude that each of the grounds of appeal raised by each appellant is unarguable.
"Neither appeal has a real prospect of success. We also consider that there are no other compelling reasons for either appeal to be heard."
The two senior judges also said that the previous ruling was "detailed and thorough".
Gatwick is the UK's second busiest airport and one of the busiest single-runway airports in the world.
The expansion will allow the site to be used for the departures of narrow-bodied planes such as Airbus A320s and Boeing 737s.
Pierre-Hugues Schmit, chief executive of London Gatwick, said the decision to approve the plans "was taken properly and lawfully".
He added: "We are also very pleased that this ruling, from senior Court of Appeal Judges, brings to an end an eight-year planning and legal process which has carefully tested and scrutinised every aspect of our expansion plans on multiple occasions.
"This final ruling means that we can now press on with the important job of bringing this exciting project to life and move forward into the design and delivery phase."
Schmit said the Northern Runway project was expected to deliver 14,000 new jobs and a £1bn boost to the economy every year.
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