My spell check is set to American English, and the other spelling appears with a red line. That's not a good start.
To verify this, I've used Google to compare the results for each spelling. I searched for
"adverti_ing" dollars, since I figured that of the time people say dollar, they mean the US dollar.
- A magazine company (it's rating is boosted by the fact "dollars" is in the URL)
- A JSOR article on advertisement in US presidential elections from 2014
- A PDF of the front page of said article
- A radio station looking for advertisers
Note that Google asks:
Did you mean: "advertise" dollars
- Second Life Online game forum
- An energy company. URL uses advertise, content uses advertize
- Probably spam. Not a good sign for spelling. (It's not even an American forum)
- A law book... from 1897
Looking at the two sets of results, advertize doesn't look very good. At best, it looks like a term that might have been acceptable 100 years ago. Advertise brings in recent, reputable publications. Google autocomplete also shows that other people use the word in searches, while searches for "advertize" are either confused over the spelling or looking for antiques.
Google NGrams agrees with this; advertise is the most popular: