While Crown properties are technically exempt from non-domestic rates, government departments pay contributions in lieu on a basis agreed with the Crown Property Unit.
Speaking at a news conference in Edinburgh, cllr Geddes described the debt as unacceptable and deeply ironic in light of Scottish office minister George Kynoch's recent comments on unpaid council tax.
'I had to sit through a lecture from Mr Kynoch on Friday about unpaid taxes,' cllr Geddes said, 'yet within days I discover that in Edinburgh alone, unpaid debt for government buildings is some £7.6m.
Debt on such a large scale has far-reaching consequences for local authorities - while large sums are outstanding, councils must borrow to make good the shortfall: 'This is an expensive procedure which ultimately rebounds on our council taxpayers with higher tax levels,' cllr Geddes said.
'We are able to pursue council tax payers who refuse to pay - and we do pursue them vigorously - but there is no action we can take against governments to claim what is a major element of out funding.'
And he stressed: 'The government, above all, should be seen to set a good example. In this instance they have palpably failed in their duty. I call on them to clear this debt immediately.'