Sir Philip, who joined RBS just months after its near collapse in autumn 2008, was setting out the bank's ambition to be number one in "all our lines of business" after declaring the mission to make it "safe and sound" was complete.
And he pledged to hold senior personnel to account if it falls short of the goals on service standards it has set.
Speaking at the ACT (Association of Corporate Treasurers) conference in Glasgow, Sir Philip said: "We want to be substantially better and regarded for our customer service right across [our business] and we aim to become the UK's most trusted bank.
"We will monitor and measure our performance on these metrics, and will publish those results.
"Your judgment, all of our customers' judgements, on the quality of service we provide will form a central part of the remuneration of our top people.
"If we don't convince you to be an enthusiastic advocate of RBS because of what we do and how we do it, then our top team will feel it in the pockets."
Sir Philip hailed the "outstanding achievement" of all RBS staff in putting the bank on an even keel, highlighting its improved position on capital, short-term wholesale funding and liquid assets, as well as shrinking its portfolio of "unwanted", non-core assets. But he conceded that recovery had taken longer than expected as a result of "legacy" issues" such as fines over the mis-selling of Payment Protection Insurance (PPI), meaning the soaring losses linked to the crash in 2008 are now only "part of the story".
Sir Philip said: "Many of the huge bills that are falling now or recently are not actually related to the financial crisis or the credit crunch, rather they are the price the industry is paying for past misconduct.
"I think this is sometimes underestimated because it has come out in dribs and drabs effectively. They are the result of poor decisions and poor behaviours around the banks, as well as some bad judgements that characterised the noughties."
Sir Philip said the bill of fines and expenses incurred by banks across the US and Europe currently runs to more than £100bn, with RBS having made total provisions of £8bn for these costs.