Golf is a sport and I agree that a round of golf should be done by walking the course rather than using a golf cart. Many people that play golf do not participate in any other sports or physical activities; so they should at least walk the golf course.
As to the question of golf being a sport or not - Golf is classified as a sport. From Wikipedia:
"Sports is an activity that is governed by a set of rules or customs and often engaged in competitively. Sports commonly refer to activities where the physical capabilities of the competitor are the sole or primary determiner of the outcome (winning or losing), but the term is also used to include activities such as mind sports (a common name for some card games and board games with little to no element of chance) and motor sports where mental acuity or equipment quality are major factors."
"The term "sport" is sometimes extended to encompass all competitive activities, regardless of the level of physical activity. Both games of skill and motor sport exhibit many of the characteristics of physical sports, such as skill, sportsmanship, and at the highest levels, even professional sponsorship associated with physical sports. Air sports, billiards, bridge, chess, motorcycle racing, and powerboating are all recognized as sports by the International Olympic Committee with their world governing bodies represented in the Association of the IOC Recognised International Sports Federations."
"Sports" comes from the old French desport meaning "leisure"
So as you can see, golf would definitely be classified as a sport.