EDIT: Perhaps the word 'mandatory' is more correct for the context.
Will some, or all games demand you to use Galaxy only and not download the games via browser?

So far, the answer has always been "no". Some games may offer certain features only if you're using Galaxy though, such as achievements.

Of course, no one knows what tomorrow holds, only the current plans for tomorrow.

Not likely, though the number of games that require Galaxy for part of the game or certain features will very likely keep increasing. Multiplayer, Achievements, Time tracking, day 1 patches, patch rollbacks etc.