Mr Johnson's predecessor Jacqui Smith formally gave the go ahead for Mr McKinnon's extradition in October 2008.
He said after a court rules there is enough evidence, a home secretary can prevent an extradition only in very specific circumstances, none of which applied in Mr McKinnon's case.
In his article, the home secretary acknowledged that it was "understandable" that many would be sympathetic to "someone who appears to be a misguided, vulnerable young man".
But Mr Johnson added that "the crimes he is accused of are far from trivial" and said Mr McKinnon "should be tried fairly for them in a court of law and in the country where the impact of those crimes were felt".
The home secretary also denied that extradition law was wrong, arguing that it was appropriate for "an age where crime is increasingly indifferent to national borders".
Glasgow-born Mr McKinnon could face 60 years or more in prison if convicted in the US.
He admits hacking by accessing 97 government computers belonging to organisations such as the US Navy and Nasa, but denies it was malicious. He also denies the allegation he caused damage costing $800,000 (£487,000).
US-UK EXTRADITION TREATY
2003 treaty, agreed in aftermath of 9/11 attacks
Offence must be punishable by one year or more in jail in both countries
US has to prove "reasonable suspicion" for extradition of a British citizen
To extradite an American from the US, British must prove "probable cause"
Since 2004, 46 people have been sent from the UK to the US for trial, and 27 from the US to the UK
Mr McKinnon has always insisted he was looking for classified documents on UFOs, which he believed the US authorities had suppressed.
He has challenged refusals by the home secretary and the director of public prosecutions (DPP) to try him in the UK.
But the DPP refused to order a UK trial, saying the bulk of the evidence was located in the US and Mr McKinnon's actions were directed against the US military infrastructure.
Harriet Harman: "Gary McKinnon should be properly looked after when he's over there"
And two judges rejected his court bid to avoid extradition, ruling that it was "a lawful and proportionate response" to his offence, even though they conceded he might find extradition and prison in the US "very difficult indeed".
Mr McKinnon has already appealed unsuccessfully to the House of Lords and the European Court of Human Rights.
But the case has led to a political row, with Tory leader David Cameron saying it raised "serious questions" about the extradition pact between the US and UK.
Liberal Democrat home affairs spokesman Chris Huhne has argued the American government would not "hang one of their citizens out to dry in the same way".
A letter has been sent to President Obama signed by 40 British MPs asking him to step in and "bring this shameful episode to an end".
Mr McKinnon's mother, Janis Sharp, has also called on President Obama to intervene.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.