Is it possible for a doctor to diagnose cirrhosis without a biopsy?

Yes. The "gold standard" of diagnosis is biopsy, but the history, physical examination, and lab tests, including liver scans and ultrasound, can strongly suggest the diagnosis to the point where a biopsy may not be necessary. New kinds of ultrasound and MRI imaging can actually determine the amount of scarring present. Most doctors will still want a biopsy to determine the severity of cirrhosis.
Yes, clincal dx. Although cirrhosis is classically a biopsy diagnosis, it is not always needed if there is sufficient clinical, lab, and radiologic data to suggest the presence of cirrhosis. For example, accumulation of fluid in the abdomen (ascites), bleeding disorders, jaundice, lab abnormalities, confusion and certain appearance of the liver on imaging may suggest cirrhosis without the need for biopsy.