A quick way to determine if a painting is American, or from somewhere else, is simply to measure it.
Standard painting sizes are not the same everywhere. The French, in particular, have a special paintings size system based on the golden rule. They have three basic format sets for portraits, landscapes, and seascapes. Seascape sizes are based directly on the golden rule. Portrait sizes are based on the golden rule x 2. Landscape sizes are based on the ratio of the square diagonal 1.414214.
Naturally, American painters worked in France, and painters from everywhere worked in Italy, and so on. The fact that a painting has a European format does not mean the artist was necessarily French or German, but like everything else, when it comes to researching paintings, it provides a hint.
A Renoir painting on an American-size canvas would be immediately suspicious.
The life of most significant painters is well documented, and it is easy to check if they ever worked in America, in England, in France, or somewhere else. With this information, it is easy to determine if a painting size is compatible with the art supplies which would have been available to an artist.
In addition to the question of standard painting sizes being different in various countries, many artists had preferences and would tend to work on canvasses or boards in a limited range of sizes.
Again, this does not provide absolute evidence for, or against authenticity, but it creates yet another presumption: if a painting matches the artist's habits, or to the contrary, is outside his usual size preferences.