Obesity isn't contagious in the same sense that fecally transmitted diseases are. "Public health" actually has a meaning beyond "big health problem".

I'm unsure what meaning you're attaching to "public health".

Here are en.Wikipedia, a UK Government site, and a training programme from the US CDC about their Public Health Prevention Service which talks about Obesity.

http://en.m.wikipedia.org/wiki/Public_health

https://www.gov.uk/government/topics/public-health

http://www.cdc.gov/phps/More.html

All three of these reputable sources specifically mention obesity as a public health problem.

What source are you using for your more restricted meaning of the term?

Sedentary working probably isn't the cause of obesity. It takes a significant amount of exercise to lose weight. There are also plenty of people in third world countries that have relatively sedentary jobs.