I'm too busy at the moment to do internet research on this, but I do know that Ukraine is (like most countries) strict about international passports. The alien spouse would need to go through the stages of applying for residency, setting up housekeeping in Ukraine, registering at a residence, and living there for years before applying to become a citizen.
I know a Ukrainian family in which a Ukrainian woman married a citizen of the Russian Federation -- even though he was already living in Ukraine for a couple of years by the time of the marriage, the hassle and paperwork he has gone through to make his status official is considerable, and moves slowly.
If I were the US citizen, I would absolutely consult a Ukrainian attorney experienced in immigration matters.
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PS - Foreigners used to "game the system" a lot by living long-term in Ukraine without the appropriate types of visas, registration, etc. A couple of years ago Ukraine tightened up their enforcement of the rules considerably. So be careful if you hear or read, "well Joe got away with such and such..."
PPS - Some Americans will tell you that "dual citizenship is illegal for Americans." This is absolutely not true, but it is a common misconception. Most countries, the USA included, don't RECOGNIZE foreign citizenship: in other words, if you are a citizen of the USA and (for example) Mongolia, under US law you are treated like any other citizen: your Mongolian citizenship makes no difference at all to the United States. But not recognizing, is not the same thing as prohibiting.
It's true that aliens who become US citizens are (at least in theory) required to renounce all foreign allegiances, but this is a complicated legal area not relevant to your question.
An ordinary person born in the USA can acquire citizenship in another country, without any problem from US law (though I would seek legal advice first if the US were at war with that country!)
My knowledge of this is based on some reading -- please do not take it as legal advice!
Under US law, as I understand it, there is no loss of rights whatever associated with dual citizenship. According to a website by a guy who seems to have researched this topic thoroughly, the one official consequence in the United States from dual citizenship is that it might prevent a person from receiving a security clearance (such as is needed for authorized access to federally classified information), or lead to revocation of clearance for a person who already has it.
On the other side (in the foreign country), one must be careful! Most of us are too old to worry about this, but holding the passport of a foreign country could make you subject to military conscription there. When you are in that country, the government could prevent from leaving the country according to whatever provisions of their laws. The US State Department could not shield you, as a citizen of that country, in the way that they do their best to intervene for US citizens having problems abroad.
It is worth keeping in mind that governments of the former Soviet countries are often extremely corrupt, and even if in theory individual rights have protections like those in the US, in practice such rights are trampled every day. As imperfect as the US system is, we can normally rely on due process, impartiality of courts, etc. Don't expect that over there! If you wind up in court in a former Soviet country, there is a serious risk that the case will already have been decided before the proceedings start.
I knew a Russian girl in the Navy stationed on the Roosevelt (CVN-71). She was a duel citizen and was denied any Rate(job) that required a security clearance. An Irish friend of mine also a duel citizen, he won the green card lottery when he was a youngster and because of that his parents and sister was also awarded the green cards. He has full benefits of both countries no restrictions except security clearance of course, he can use both his U.S. passport or Irish passport as he chooses.
I also researched how I would be able to get Russian citizenship if I were to marry there and the process is probably similar to Ukraine. You have to get the appropriate paper work from the U.S. embassy saying you are not married, and if you are divorced the paperwork showing that you are indeed legally divorced. Then you have to have a Russian translator change the documents into Russian and get it notarized. The wife will also have to provide the same paper (though much easier for her to do) and you have to show all the legal documentations of your marriage in Russia/Ukraine, pay all the fees and wait. You will then be granted a temporary residence to live and work in the country for a matter time, a year maybe longer then you can apply to the same agency to full permanent resident and receive passports and citizenship.