Not quite! This is making a statement about conditional probability, but you got the condition reversed: people who are depressed are more likely to be "engaged fully in the 'web' experience".
To find out the chances that a heavy web user is depressed, you have to also consider the other things that can cause someone to use the web heavily. There's a really slick theorem for calculating this:
Wow... I have recently taken courses that taught Baye's Theorem, and even more recently read a long, well-written piece about how even qualified people often misinterpret statistics (the way your parent did which is not congruent with Baye's theorem), and yet I still interpreted the stat in the same, incorrect way.
That seems to be one of the major laments in Thinking, Fast and Slow (Kahneman). Us humans, even when we're 'ready for it', tend to be remarkably inflexible in adopting concepts which conflict with our basic or social intuitions...
http://en.wikipedia.org/wiki/Conditional_probability
To find out the chances that a heavy web user is depressed, you have to also consider the other things that can cause someone to use the web heavily. There's a really slick theorem for calculating this:
http://en.wikipedia.org/wiki/Bayes_theorem