NFL Films coined it, you can thank them and give proper credit where credit is due.

"Bob Ryan, now Vice President and editor-in-chief of NFL Films, coined this for the Cowboys while preparing and editing the team’s 1978 season highlight film.[6] He was quoted as saying:

I wanted to come up with a different twist on their team highlight film. I noticed then, and had noticed earlier, that wherever the Cowboys played, you saw people in the stands with Cowboys jerseys and hats and pennants. Plus, they were always the national game on television.

Ryan told the NFL Network for their Top Ten Nicknames show:

I saw all these fans in away stadiums. Hey, they're the most popular team in the country. How can I use that? Why don't we call them "America's Team"? [1]


Here is another reference: https://247sports.com/nfl/dallas-cowboys/Article/How-the-Dallas-Cowboys-became-Americas-Team-119561925/
During the Cowboys' first game of the 1979 season, a nationally televised game against the St. Louis Cardinals (Dallas won 22-21), the television announcer for CBS introduced the Cowboys as America’s Team and the name stuck.