He watched the boat ____ down the river.

Watch-এর পর verb-এর base form অথবা verb+ing দুটিই বসতে পারে। একটি simple sentence-এ দুটি main verb ব্যবহার করা যায় না, তাই verb-এর সাথে ing যোগ করতে হয়।

তাই floating সঠিক উত্তর।
Complete sentence: He watched the boat floating down the river.

Collins Dictionary: If you watch someone or something, you look at them, usually for a period of time, and pay attention to what is happening.
Example: Chris watched him sipping his brandy.

Longman Dictionary: watch somebody/something do/doing something
Example: I watched him go, then went home.