No. Any version of any browser that can download the installer should work.
Installing Avast might install the latest version of Chrome. There might be a checkbox (so can say no), but maybe there isn't. Then you can uninstall it via Control Panel, if you don't want it.
Kees
Hello friends,
Downloading the latest version of avast also need latest version of Firefox-internet explorer?

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic