I can't seem to put weight on my wrist (push up to surf). Do I have carpal fracture?

Need an xray. The only sure way to know if you have a fracture is to obtain an xray of your wrist. Other causes can be ligament sprains, ganglion cysts, tendinitis. If it persists after rest, ice and maybe anti-inflammatory medicines, then best to have it evaluated by a physician.
Perhaps. It is hard to know without examining your wrist and without x-rays. Did you have a specific trauma to your wrist, or is this a chronic problem with no specific injury? If it is a chronic issue without antecedent trauma, a fracture is less likely though still possible. Either way, best to have it examined by a physician, as an undiagnosed bone fracture can have severe long term consequences.