The service would have been the first ever domestic service in Britain by the Irish airline.
The valuable slots – they rarely become available at capacity-constrained Heathrow – had been surrendered by British Airways as one of the conditions laid down by the European Commission for it being allowed to buy BMI earlier this year.
The service between Heathrow and Edinburgh will compete directly with British Airways, which is part of the Willie Walsh-led IAG group that also includes Spain's Iberia airline.
Virgin said it plans to launch the service to Edinburgh by March of next year – the same timetable Aer Lingus had hoped to achieve. Aer Lingus was informed last week by the European Commission that it had been ranked second in the competition to procure the Heathrow slots.
Aer Lingus chief executive Christoph Mueller recently said that the airline would have difficulty growing organically and would consider small asset purchases in order to help it expand.