No. "To watch" means something more specific than "to look at". It means to observe over a period of time, to observe a process. You can look at a book as easily as you can look at a movie, but you cannot watch a book in the same sense that you can watch a movie.
We watched the government and business community cope with the ongoing crisis.
This is a perfectly ordinary sentence. We can divide its one clause into four constituents: a subject, a finite verb, a direct object and an object complement. This sense of the verb "to watch" does license bare infinitive object complements, and the phrase "[to] cope with the ongoing crisis" is a suitable infinitive phrase. As modified by its complement, this direct object represents a process with a duration. It makes sense as the object of the verb "watched".
We watched that the government and business community cope with the ongoing crisis.
There is something wrong with this sentence. It has a perfectly ordinary (and somewhat more complicated) structure, but it doesn't quite make sense. At least, it doesn't make the kind of sense that you intend.
The intended sense of the verb "to watch" does not require an object complement. It does require a direct object. We have a direct object in the subordinate clause "that the government and business community cope with the ongoing crisis". In this case, "cope" must be finite. The "that" introduces a complete clause and allows that clause to represent a fact.
And there's the problem. A fact, like a book, isn't a process -- even when the fact is that a process occurs. The fact itself lacks an inherent duration. It simply exists. We cannot watch a fact in the same sense that we can watch a movie.
For this sentence to make sense, we would need to find some sense of the verb "to watch" that licenses a different semantic role. These senses can be found, such as the religious sense of to hold vigil or to pray throughout the night which can license a purpose rather than a process.
Alternately, we can replace "watched" with a verb that licenses a more appropriate semantic role, one that does not require an inherent duration. Verbs like "saw", "observed" and "noticed" have fewer semantic restrictions regarding their direct objects.