There are no marks on the back or sides of the case. This belonged to my grandmother (born 1928). I do not have the tools to open the case to look at the movement. I have no other information about the watch and do not know the date of manufacture, or when it was purchased.

Good looking watch and a nice heirloom. We have only been able to determine this is a mens watch, and have not yet found an ad to identify it. There are a couple like it in the database categorized as Unknown seen here, here and here.
I suspect your watch dates to the late 1920's like the three I reference. It's pure speculation, but I suspect your grandmother may have inherited it from her father. If the watch dates 1928 your grandmother's father may have received it or purchased it in celebration of her birth. The only way to tell for sure what the date is, is to see the serial number on the inside of the hinged case back. If only it could talk.....
Unknown.
I'll change the year above to 1928 as it is not a 1940 - wrong style. Here is an ad from 1929 with your watch shown upper right below Miss America. Unfortunately, no name is given.
Perfect! The shield on the movement next to the serial no. is the mark for 1929. The serial number inside the back starts with a "9", also indicating 1929. The Stamped date of 1927 is the year of the patent for the celluloid dust cover.
Your watch is a 1929 (for now) Unknown.
Date updated to 1929, movement, case s/n's and movement type added above.