Can a person coming to the U.S. on a Visa Waiver get a Green Card through Marriage?

The answer is: YES

In a brief filed in Bradley v. Holder Case #10-397, the Solicitor General said:            

In general the Visa Waiver Program aliens are excepted from eligibility to seek adjustment of status but those who qualify as immediate relatives fall within an exception to the exception. See 8 U.S.C. 1255(c)(4)

A spouse is an immediate relative.  However, you should note that it is still considered immigration fraud to come to the USA on a tourist visa or the VWP with the intent to immigrate.  But if your intent was truly to visit and then you later decided to marry a U.S. citizen, you would be able to adjust your status and get your green card.  However, if you are currently in removal proceedings you probably will not get to stay just because you married a U.S. citizen and applied to adjust your status.

In an engagement with the American Immigration Lawyers Association on April 7, 2011 the USCIS stated:

All field offices have been instructed to adjudicate I-485 applications filed by individuals who last entered the U.S. under the Visa Waiver Program (VWP) and overstayed on their merits UNLESS the potential beneficiary is the subject of an INA section 217 removal order.  Additionally field offices have been instructed to hold in abeyance all VWP adjustment applications for potential beneficiaries who have been ordered removed under INA section 217.