Greenwood, a try-scorer in England's crucial 25-6 Pool C win over the Springboks on Saturday, will miss next week's clash with Samoa, but he should be able to play against Uruguay on Sunday, November 2.
Woodward said in a statement: "Will is flying home for a week in order to be with his wife Caro, who is experiencing difficulties with her second pregnancy. On behalf of the squad, we would appreciate it if Will and Caro could be given complete privacy at this time.
"I will not be flying out any direct replacement for Will or bringing anybody out on standby as we have sufficient cover in the squad. Will is expected to return on October 25 to prepare for the Uruguay fixture.
"We are formally notifying the Rugby World Cup organisers this morning and had planned to make the announcement on Sunday afternoon.
"However, due to a leak to a Sunday newspaper, the England management is making the announcement earlier than expected."
Woodward also went on to pay tribute to Greenwood, who had kept the news quiet from his team-mates until after the game against SA.
"I have to pay great tribute to Will. I have known about this all week and my wife Jayne went to visit Caro earlier this week," said England's coach.
"They tragically lost their first baby Freddie after 21 weeks of the pregnancy and obviously there a few problems with this second pregnancy. She is in hospital and will be in hospital for the next month or the remainder of the pregnancy.
"It has been a tough week for Will and he has handled it brilliantly and I have to say that Caro has been fantastic too. She wanted him to stay, but it is critical now that we get him back on a plane so he can be with her. And all our wishes are with both of them and I hope that it all goes well.
"Will wanted to keep it quiet this week and not worry the players. He's been outstanding and considering what happened to his first child it's unbelievable. I've been speaking to him daily, hourly almost, and he wanted to play."