Antony Blinken said the US was focused on actions such as the treatment of opposition figure Alexei Navalny, the Solar Winds hack and election interference.
“We would prefer a more stable and predictable relationship,” he said.
Mr Blinken was in the UK for a meeting of foreign ministers of the G7 group of industrialised nations.
A statement issued after the two-day talks criticised Moscow for its “irresponsible and destabilising behaviour”, particularly against Ukraine, and for cyber-attacks.
Mr Blinken echoed those sentiments on Thursday, offering support to Ukraine during talks in Kyiv with its President Volodymyr Zelensky, who said his country was under threat from Russian troops massed near its border.
Back in February US President Joe Biden said, in more headline-grabbing terms, that he had made it clear to President Vladimir Putin “that the days of the United States rolling over in the face of Russia’s aggressive actions… are over.”
His predecessor Donald Trump had at times seemed to avoid criticising the Russian leader.
Mr Blinken told BBC Radio 4’s Today Programme that the Biden administration was not trying to hold China back, stressing that the US was in favour of upholding a rules-based international system.
He said countries needed to look very carefully to see if China was investing in their strategic assets.