No, that option has been abolished in the 1920s, and even back then it only ever applied to women. Today, nobody can become a US citizen through a spouse.
Edit: I misread the question. You did ask about a Green Card. The rest of the answer of course remains correct.
What he may be able to do is apply for an immigrant visa based on the marriage. That would allow him to move to the US and live and work in the US, but he would remain a UK citizen (or at least, not a US citizen).
That requires exactly the same things as any other marriage-based immigration case. In particular, if he is inadmissible, he can't get it. That could be caused by something as simple as DUIs, or possibly an arrest if he was a kid, or various other things even if they were legal in the country where he did them. US immigration law does not recognize things like sealed records or expunged records, either.
Also, as others have already pointed out, if he renounced his citizenship to avoid US taxes, he may be permanently banned from the US (even as a tourist).
One other pitfall: from the minute he starts the paperwork until he actually has the visa in hand, he will have serious problems traveling to the US, and could well get deported if he tries (which in turn would probably cause the immigrant visa to automatically get denied). In particular, DO NOT try to use the visa waiver. If you have to travel to the US in the interim, apply for a tourist visa at a US consulate and hope that it will get approved.
Once he has the immigrant visa, becoming a citizen has almost nothing to do with his marriage; except for the mandatory waiting period, it is the same procedure for everybody.
- Both he and his wife have to physically move back to the US within six months (the children would probably move, too, obviously).
- They both have to reside in the US for three years from the date they arrived on US soil. If they get divorced, the time becomes five years.
- Shortly after the three (or five) years are up (90 days before), he can file an application for citizenship.
- One that is approved (should be pretty quick, usually only a year or two), he will be invited for a citizenship interview where he will have to take the famous citizenship test. He'll likely ace the English language part, obviously, but he better brush up on his civics. Can you name all 13 original colonies?
- Once he passes the citizenship test, he will have the citizenship oath scheduled. He will become a US citizen the day he takes the oath. Hint: usually, they accept passport applications immediately after the ceremony, so bring that paperwork if you want to travel soon.
Eligibility for citizenship wouldn't be based on his marriage, but rather based on the fact that he has been a bona fide US immigrant for the required period. The process can take another year or two.
Once he has been sworn in as a US citizen, he could then move back to the UK.
He would get a few benefits from the marriage:
Getting an immigrant visa is a very quick process; he should have it in hand within two or three years, sometimes even less than one year. Compared with other people, that is lightning speed for immigration purposes. And of course, the vast majority of the world's population would not even be eligible to start the process in the first place.
As long as he remains married, the wait until he can apply for citizenship is reduced to three years.
There is an exception to the three-year rule: if his wife works for the US government abroad, or for certain international corporations, then it is enough to get his Green Card (i.e., get the immigrant visa, and set foot on US soil to convert it to a Green Card), and he can then apply for US citizenship immediately.
All in all, don't do this without an immigration lawyer. You don't want to mess it up, or else you may end up not even being allowed to visit the US ever again.
