No. The illegal alien must return to their country of citizenship. NO way to change their status, or get any type of visa or green card, even through marriage if the person is not legally present. They return to their own country, and will be barred from reentry (most commonly 10 years, since they typically have been in US illegally for a year or more, unless they have committed any crime, been deported previously, etc.).
They can arrange their US spouse's immigration to their country once they are back there. If spouses do not live together, it will become very difficult to prove a viable, ongoing & committed marital relationship to attempt sponsoring them for immigration after they complete their bar to reentry. Waivers of inadmissibility are becoming tougher & tougher to get, and most people simply do not even qualify to apply. Sure, you can hire an attorney, at costs of up to $25,000, to no avail. Plan on moving to your illegal alien spouse's country, and perhaps permanently, not just for the duration of their bar. Their record of immigration violations will always be against them.