Stagecoach says it is "extremely concerned" after the Department for Transport (DfT) barred it from three UK rail franchise bids.
The DfT says the bids for the East Midlands, South Eastern and West Coast franchises were "non-compliant" because they did not meet pensions rules.
Stagecoach had "repeatedly ignored established rules", the DfT said.
Mr Griffiths said in a statement: "We are extremely concerned at both the DfT's decision and its timing. The department has had full knowledge of these bids for a lengthy period and we are seeking an urgent meeting to discuss our significant concerns."
Bidders for the franchises have been asked to bear full long-term funding risk on relevant sections of the Railways Pension Scheme, Stagecoach said. The Pensions Regulator has estimated the UK rail industry needs an additional £5-6bn to plug the pensions shortfall, and the company said it was being asked to take on risks it "cannot control and manage".
Mr Griffiths said: "Forcing rail companies to take these risks could lead to the failure of more rail franchises and cannot be in the best long-term interests of either customers, employees, taxpayers or the investors the railway needs for it to prosper. It was more evidence that the current franchising model is not fit for purpose" and "further damages the already fragile investor confidence in the UK rail market".
Stagecoach had bid independently for the East Midlands franchise, had intended to partner with Alstom for the South Eastern operations, and was jointly bidding for the West Coast Partnership with Virgin and SNCF. A DfT spokesman said that other bidders had met its requirements, and also announced that the East Midlands franchise had now been awarded to Abellio "after they presented a strong, compliant bid". He said: "Stagecoach is an experienced bidder and fully aware of the rules of franchise competitions. It is regrettable that they submitted non-compliant bids for all current competitions which breached established rules and, in doing so, they are responsible for their own disqualification. Stagecoach chose to propose significant changes to the commercial terms for the East Midlands, West Coast Partnership and South Eastern contracts, leading to bids which proposed a significantly different deal to the ones on offer.
While Stagecoach has played an important role in the UK railways industry, "it is entirely for Stagecoach and their bidding partners to explain why they decided to repeatedly ignore established rules by rejecting the commercial terms on offer".
Stagecoach, which also has a huge bus division, currently operates the East Midlands rail franchise between London St Pancras International and destinations including Leicester, Derby, Sheffield, Nottingham, Manchester and Liverpool.
The company has a 49% share in Virgin Rail and also runs the Sheffield Supertram.
Stagecoach's East Coast franchise was renationalised last year following poor performance and mounting losses.
As a result of the above.....................
Virgin Trains 'could disappear' after franchise bar
Virgin boss Sir Richard Branson says his train business could disappear from the UK after its partner Stagecoach was barred from three rail franchise bids. Sir Richard, whose Virgin Trains is 49% owned by Stagecoach, said he was "devastated" by the disqualification.
The Department for Transport (DfT) disallowed the bids because they did not meet pensions rules.
Virgin was bidding to renew the West Coast franchise in partnership with Stagecoach and France's SNCF.
In a blog on Virgin's corporate website, Sir Richard said Virgin Trains "could be gone from the UK in November". "We're baffled why the DfT did not tell us that we would be disqualified or even discuss the issue - they have known about this qualification in our bid on pensions for months," he wrote. "The pensions regulator has warned that more cash will be needed in the future, but no one knows how big that bill might eventually be and no responsible company could take that risk with pensions. We can't accept a risk we can't manage - this would have been reckless. This is an industry-wide issue and forcing rail companies to take these risks could lead to the failure of more rail franchises."
The deadline for bids is now closed, so any reopening of the process seems unlikely. There are thought to be two remaining bids left for the West Coast franchise.