Not without a ridiculous amount of effort.
Original serial mice are relatively dumb. They generate a stream of serial data in the form of bursts of characters that can be received by an RS-232 serial port and interpreted by the driver.
USB is a bidirectional protocol that requires intelligence at both ends and does a lot more than just send updates when the mouse is moved.
During the changeover period from serial to PS/2 to USB some mice were made that could be used with simple adapters. Specifically, some PS/2 mice could be used with an RS-232 port. This was when people assumed that PS/2 was the way of the future, and people would eventually upgrade their PCs and be able to throw away the adapter. I think some USB mice can also be used with PS/2 ports, with the right adapter.
But original serial port mice didn't know about these new interfaces, and can't be used with them. The only way would be to make a smart adapter that responds to the serial data from the mouse, figures out how the mouse is being moved, and provides a USB slave implementation that can talk to the USB port on the PC. This would require a microcontroller and a lot of programming.
Short answer, buy a modern mouse. The new mice may have no balls, but they also don't need to be cleaned