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The defeat means it is now mathematically impossible for sixth-placed United to qualify for the Champions League, as they sit five points adrift of fourth-placed Arsenal with one game left to play.
Rangnick, who will make way at United for Erik ten Hag at the end of the season, has said he will also stay on in a consultancy role at the Premier League club.
Greenwood remains suspended by Manchester United until further notice.
Ronaldo's 62nd-minute strike was his 17th league goal this season with only Mohamed Salah of Liverpool having found the target more often.
The Portuguese superstar's goal wasn't enough for United as Arsenal ran out 3-1 winners at the Emirates.
United has been led by interim managers Michael Carrick and currently Ralf Rangnick since Ole Gunnar Solskjaer was fired in November.
Cristiano Ronaldo received an outpouring of solidarity from Manchester United fans and their Liverpool counterparts during a Premier League match he missed after the death of one of his new-born twins.
After a 5-0 win at Old Trafford in October, this was another humiliation of United by Juergen Klopp's team, who were barely threatened by their lifeless opponents.
The Portugal international is expected to train today.
Merseyside police said on Sunday they were reviewing CCTV footage to establish if Ronaldo committed an offence by knockingthe mobile phone out of the supporter's hand.