The Republican Party did not exist in 1850.
Technically, I think you're right. Who might their predecessor party have been? A little history would be helpful but I suppose if anyone's interested they can dig it out themselves...
I believe people in the Democratic Party where referred to as Republicans at that time.
The Compromise of 1850 led to the way of a much stronger Fugitive Slave Act that led to the specter of slaveowners traveling north to reclaim runaways on free soil with the active assistance of federal officers. Then Stephen Douglas introduced the Kansas-Nebraska Act to allow "popular sovereignty" i.e. local option on the issue of slavery, which effectively repealed the Missouri Compromise and let to the next specter of slavery expanding into the northern parts of the Louisiana Purchase. This led to "Bloody Kansas" in which free soilers fought with pro-slavers to control the territory. These acts outraged free soilers across the north and led to the demise of the Whigs, which had been the conservative opposition to the Democrats, but could never reconcile it's Northern and Southern wings.
"Anti-Nebraska" conventions in Ripon, Wisconsin and Jackson Michigan led to the birth of the Republican Party. It was composed of anti-slavery Whigs, free soil Democrats and some third party folks from the free soil and "know nothing" parties.