Not directly--with only 32kb of program space in the Arduino Uno you can just barely get a wifi stack in (see the CC3000 chip), but there's no room left for SSL or anything secure.
A better option is something like the Arduino Yun which has two processors, a small Atmel microcontroller and an Atheros system on a chip running OpenWRT linux. The linux processor can act as a bridge to let the microcontroller communicate with the internet pretty easily.
A better option is something like the Arduino Yun which has two processors, a small Atmel microcontroller and an Atheros system on a chip running OpenWRT linux. The linux processor can act as a bridge to let the microcontroller communicate with the internet pretty easily.