Arthur Gumbley - known as Bob - was attacked at his house in Endwood Drive, Sutton Coldfield, between 7pm and 9pm on 21 November, and jewellery, cash and a purse belonging to his late wife was stolen.
He was found lying on the floor and covered in blood, having been punched in the face and kicked while on the ground, and was taken by ambulance to Good Hope Hospital - where he died on Monday.
Two men aged 42 and 34, and two women aged 33 and 29, have been arrested on suspicion of the 87-year-old's murder and remain in custody, Staffordshire Police said.
Detective Inspector Dan Ison said: "This was an horrific and despicable attack on a defenceless elderly man in his own home, which has now resulted in his untimely death and a murder investigation.
"I am appealing for the support of the public to assist us in bringing those responsible to justice. Therefore, I make a plea to any person who may have any information, or who may be a witness to this terrible incident, to please make contact with us.
"Crimes of this nature cannot go unpunished, and I want to reassure Arthur's family that every effort is now being made to bring those responsible to justice. They are being supported by specially trained officers, and I would ask that their privacy is respected at this sad time to allow them to grieve."
:: Anyone who can help police with the investigation is urged to call 101, quoting incident number 828 of 21 November.