"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," Johnson added.
In a joint statement last week, Britain, France and Germany said the Iran deal was "working and delivering on its goal" and said they "deeply regret" the reimposition of US sanctions.
On Sunday, Johnson seemingly disregarded the move, urging British businesses directly to cut ties with Iran.
"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," Johnson wrote in the Telegraph.
"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."
Johnson's comments come less than a week after the Trump administration on Tuesday reimposed a raft of sanctions on Iran that affect, among other things, the purchase or acquisition of US dollars by the Iranian government, the country's auto industry and trade in gold or precious metals.
Another phase of US sanctions will be reimposed in November and will target Iran's crucial oil industry.
Shortly after the sanctions snapped back into place last week, Trump warned in a tweet that countries doing business with Iran would "NOT be doing business with the United States."
Trump has long been a fierce critic of the nuclear deal, agreed under his predecessor, calling it "insane" and maintaining that, even with the current restrictions in place, Iran continues to pose a threat to the US.
Joe Kaeser, the chief executive of German industrial conglomerate Siemens, summed up the harsh realities of US economic influence over European business when he told CNN in May that the company would stop all new deals in Iran, following Trump's withdrawal from the nuclear deal.
"There is a primacy of the (US) political system. If that primacy is 'This is what you are going to do,' then that is exactly what we are going to do. We are a global company. We have interest and values and we have to balance both," said Kaeser.