In National Parks they can definitely arrest you. I know a number of people who've been arrested and spent a night in the Yosemite Valley jail for shit like being drunk and disorderly in public.

At the state and local level it probably depends on where you live. Out here in California the rangers carry guns, write tickets for shit like speeding (on mnt bikes LOL), etc. They are considered to be law enforcement. That being said, I know from personal experience that most are extremely lenient and will often look the other way. people become rangers to be part of nature, people become cops to enforce the law.