From this thread, it looks like the reason your Greet option is grayed out is precisely because your friend sent you Greetings that you acknowledged.
Yup! You auto-greet the first friend that greets you, or more specifically, the one whose character you first click on. That's why the Greet button is grayed out.
You can greet once a day only.
On a new day, you probably clicked on someone else's greet to you, which automatically makes you do your greet too. So the buttons is basically perma darkened unless you go out of your way to avoid other peoples greetings to you.
I have since been able to replicate this and it appears that you are informed that you are sending greetings when you accepted a friend's greeting:
If you haven't done this yet on a day, then you should just be able to select Greet like this: