My spell check is set to American English, and the z 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 (its 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. (It's not even an American forum) Not a good sign for spelling.
- A law book... from 1897
(These results may be different for you — they are mostly different for me after several years too. In fact, this question is the number one result when I do the second search. But it still works as evidence towards the same conclusion.)
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: