American Democrats are more right leaning than the Liberals are. We were one of the first countries to legalize gay marriage during the last Liberal term under Paul Martin, and marijuana was well on it's way to being decriminalized/legalized until the conservatives took over. They also put in heavy banking regulations that saved our economy when things went to shit in '08 (something Harper opposed, but took credit for later). That's pretty much the opposite of what the Democrats did under Bill Clinton and repealing Glass-Steagall
The closest thing we have to the democratic party is actually the conservatives. When Harper took over they were pretty similar in most aspects, though he's taken them more to the right the longer he's been in power