The probe into RBS's sale of financial products from 2005 to 2007 was being carried out by the US Justice Department.
Chancellor Philip Hammond said last year that he was reluctant to sell shares while the DoJ probe had still to be resolved.
The DoJ said further details must be negotiated before a final deal is done.
McEwan said: "Today's announcement is a milestone moment for the bank.
"Removing the uncertainty over the scale of this settlement means that the investment case for this bank is much clearer."
He added: "Reaching this settlement in principle with the US Department of Justice will, when finalised, allows us to deal with this significant remaining legacy issue and is the price we have to pay for the global ambitions pursued by this bank before the crisis."
RBS said about £2.6b of the penalty would be covered by money already set aside.
The deal moves the bank a step closer to shake off government ownership. The government paid £45.5b to bail out the bank after the financial crisis.
Last month RBS said it is to close up to 275 branches as it fights to cut costs.
Ross McEwan said the axe will fall on English and Welsh outlets run under its Royal Bank of Scotland brand.
Many of these are across the street from branches operated under RBS's NatWest brand, and customers will be able to use those instead.
The cuts will be on top of 180 closures already announced by RBS, triggering fresh fears for the future of High Street banking.
Campaigners warn that small businesses, the elderly and vulnerable people suffer most when branches are shut down because they are deprived of access to basic financial services.
RBS was previously ordered to sell the 275 branches to satisfy competition regulators as a condition of its £46b UK Government bailout during the financial crisis.
It spent the best part of a decade trying to split them off as a new bank called Williams & Glyn, but this proved too complicated.
Instead RBS is launching a £775 million fund to help rivals improve their business banking to satisfy the demands of competition watchdogs.
It means the lender is finally able to close branches, saving millions of pounds on staff and rent.