Maggie's Farm

Not quite. Humans brought dogs with them when they migrated into the Americas across the Bering Land Bridge. Most of them seem to have been kept either as working dogs (hunting, pulling sleds or travois) or as meat animals. But the Aztecs and Inca bred some down to lap-dog size and buried them with high-status people, which would seem to indicate they were pets. These breeds are now the Chihuahua and Peruvian hairless dogs (although now they have significant post-contact dog admixture as well).

Sadly, most of the pre-contact native dogs were also wiped out after contact, due to introduced canine diseases. There are a few dog breeds from that time that have survived, like the Carolina Dog (or Carolina/Dixie Dingo, or Yellow Dog).

We don't know a whole lot about how the Native Americans viewed their animals, such as the guinea pigs that Incas kept. I'd say that, on the whole, human beings didn't really start treating animals as pets (i.e. having an emotional connection to them) until quite recently. If you kept dogs or cats, it was for work, not for companionship; you didn't bond with them, any more than you bonded with cattle or pigs. (My opinion: once people started having fewer kids, they displaced their maternal/paternal instincts onto animals instead.)