Stagecoach said: “We do not believe the granting of these services within a franchised system and without a level playing field is in the best interests of passengers, taxpayers or communities.”
The company said it was reviewing its options, but it cannot appeal against the decision. Its shares fell 1% to 250p.
Four years ago, Virgin challenged a decision to award the West Coast rail franchise to FirstGroup, which led to the contract award being scrapped.
Virgin Trains said: “We have concerns over the granting of the open access services to Edinburgh, which will have a significant impact on the money received by taxpayers, since open access operators take fares but make no contribution to Treasury funds.”
Stagecoach is paying £3.3 billion to the Treasury over the eight-year life of its franchise on the East Coast, which began last year.
The track access charge regime is under review and new charges could be levied on open access operators from 2019, which would make it harder for First to offer passengers ultra-low fares.
Virgin Trains East Coast has won permission from the rail regulator to run additional services, which will run between London and Edinburgh, Harrogate, Lincoln and Middlesbrough.
FirstGroup’s services — five a day — will run from May 2021 between London and Edinburgh via Stevenage, Newcastle and Morpeth.
Its chief executive, Tim O’Toole, said: “We will offer genuinely low fares at half the average price of today. Our brand new trains will be cheaper than other rail services, greener than the plane, quicker than the coach and will get passengers from London to Edinburgh earlier than they can arrive today.”
