Alexander Paton, 68, was accused of marrying the women at a registry office in Glenrothes and a church in Greenock over seven years.
Paton, of Kirkcaldy, Fife, was due to go on trial at the town's sheriff court yesterday, but the case was dropped after prosecutors said they would "not proceed".
Paton's lawyer earlier said the case could be "resolved" once divorce records from the US were sent to Scotland.
It was alleged Paton married Filipino Perla Montilla, 65, in Glenrothes in 2006, while married to another woman, Dorothy Campbell. It was also claimed that while married to Perla, he married Romanian Judit Gherghiteanu, 67, in 2007.
A third charge alleged he illegally married Margaret Nicol, 66, from Fife, at a church in Greenock in 2012.
He had denied the charges.