In spite of the 35.4p surge to 364.2p for the share price, which added more than £4 billion to RBS's market capitalisation to take it to £41.6 billion, the stock remains well adrift of the 502p break-even price for the taxpayer's 81 per cent stake.
The bank was scheduled to put out its numbers on Friday but decided to release them early because they were more robust than the market had anticipated.
Although income in the first six months of the year actually fell, from £10.6 billion to just short of £10 billion, pre-tax profits were hiked 93 per cent from £1.37 billion to £2.65 billion.
RBS was helped by a major fall in impairment charges for bad loans, from £2.15 billion to £269 million, while also booking a £191 million gain on the disposal of the remainder of its stake in Direct Line Insurance.
While Mr McEwan stated his belief that RBS is a "fundamentally stronger bank" than it has been he was cautious on the prospects for continued progress. He said: "I'm pleased we've had two good quarters, but no one should get ahead of themselves here - there are bumps in the road ahead of us."
Issues relating to possible foreign exchange manipulation, mortgage-backed securities in the United States and ongoing redress to personal and business customers in the UK related to mis-sold payment protection insurance (PPI) and complex loans were cited as factors which will impact on profits.
RBS took £150 million of provisions in the period for PPI to take its total hit from the scandal to £3.25 billion.
There was also a further £100 million set aside for compensation to companies which took interest rate swap loans taking RBS's provisions for that to £1.3 billion.
Total operating expenses were reduced from £7.75 billion to £7.1 billion as a result of lower litigation and conduct costs, £250 million against £620 million.
However the bank's continuing restructuring push led to £514 million of cost in the period to June 30, compared to £271 million the previous year.
There was also a positive update on the lender's internal bad bank which is now expected to cost between £2.5 billion and £3 billion, below initial predictions of up to £4.5 billion.
As well as that RBS was able to recover some of the loan losses from there after selling assets at better prices than anticipated.
Chairman Sir Philip Hampton said: "We have not been much overburdened with good results in recent years so these are pleasing. But we should not kid ourselves they are entirely satisfactory. We still have a lot more still to do."
Mr McEwan and RBS's chief financial officer Ewen Stevenson, recently joined from Credit Suisse, both expect loan impairments to be higher in the second half of the year.
Mr Stevenson said: "We don't expect to see continuation of those trends into the second half."
Mr McEwan said the proposed divestment of the Williams & Glyn branches is unlikely until at least 2017 and said a complex operation to remove its systems from RBS is likely to run until 2016. There was no specific detail on Ulster Bank, following speculation a private equity-backed management buyout of the Republic of Ireland part of the business was moving closer. Mr McEwan said a number of options were being considered.
Mr McEwan said as branch usage continues to decline the bank will move forward in its closure programmes.
He pointed out it has struck a deal with post offices to allow customers to perform simple banking transactions there while also investing in more mobile vans to service remote areas of Scotland.
The chief executive also confirmed RBS is investing heavily in its digital services.
On the Scottish independence referendum, Mr McEwan said it was a decision for the people to make and RBS would not enter the debate.