Mr Johnson’s comments escalate the tensions over what is the first test of the special relationship between the US and UK since Mr Trump’s successful visit to the UK last month.
Britain negotiated the original Iran deal in 2015 with France, Germany, China, Russia, US, in a process coordinated by the European Union.
It saw the regime agree to curb its nuclear programme in return for the lifting of sanctions.
But Mr Trump pulled America out of the deal, which was struck by his predecessor Barack Obama, three months ago because Mr Trump believes it is not stopping Iran from meddling the Middle East. This week hard-hitting economic sanctions were reimposed.
In his article Mr Johnson writes: “America is turning up the pressure and we want the UK by our side. It is time to move on from the flawed 2015 deal.
“We are asking global Britain to use its considerable diplomatic power and influence and join us as we lead a concerted global effort towards a genuinely comprehensive agreement.
“Together, we can help bring about the peace and prosperity in Iran that the whole world wants to see.”
Mr Johnson even goes over the head of Whitehall and urges British businesses directly to cut ties with Iran. The UK’s trade with Iran was £365million in 2016.
He writes: “The President has been explicit: any businesses which put their own commercial interests in Iran ahead of the global good will risk serious consequences for their trade with the United States.
“Only by presenting a united front can we exert the maximum possible pressure on the Iranian regime and get them to finally change course and put an end to their malign and reckless activities both at home and abroad.”