Yes, if I was a D, I'd stay quiet for now.
If you investigated a C, yes. I'd stay quiet in that situation too.
But you'd be a fool if you investigated a G and still stay quiet. You have 100% guaranteed protection for 2 days. That means guaranteed three investigations. But the chance the Gs to kill both jailers and the bodyguard in the first days is too small so you'd probably be protected for 3 days and get at least 4 investigations. If you don't reveal yourself then there is always a chance to be whacked today already.
Of course, it could be tricky to convince the others that you are the D, but I convinced them and Hoori convinced them. Rarely someone is foolish enough to act as the D on day 1.
In any case, the D can do whatever he thinks is right. I'm telling him the potential consequences of his actions in case he investigated a G. The Gs already know that the D got one of them if the D investigated a G today.
But chances are bigger that the D got a C today so everything we're saying now is futile. If the D investigated a C then he MUST stay quiet and hope that the Gs won't get a lucky guess and whack him.