Anuj Bidve, a 23-year-old postgraduate student, was shot in the head at close range early on Monday in Salford as he walked into Manchester city centre with Indian friends.
The murder is being treated by detectives as a "hate crime" which may have been racially motivated.
Greater Manchester Police's Assistant Chief Constable Dawn Copley confirmed that Subhash Bidve, the victim's father, read a post on the social networking site before officers were able to contact him by more conventional means.
She said: "That is not the way anyone should have to find out something so devastating and we completely understand how upset the family are.
"Social networking is instantaneous and we have no control over when and what people post on such sites, but no one should hear such tragic news in this way."
Earlier, the father told BBC radio: "Nobody official from the UK government or consulate or the Indian government called us and told us about this.
"I am really surprised because they confiscated his phone and must have known his father's or mother's number.
"They could have called us and told us what had happened to him."
Police said a 16-year-old boy and two 17-year-old boys arrested over the murder have been released on bail.
Two men aged 19 and 20 remain in custody.
Bidve, from the western Indian city of Pune, was studying micro-electronics at nearby Lancaster University and had been on a short break in Manchester with eight Indian friends when he was shot.
Police said the students were walking from their hotel towards the city centre when they became aware of two men on the other side of the street.
A white male walked across the road and after a brief conversation with the victim produced a gun and shot him at close range, before running off. Bidve died in hospital a short time later.
Police have not revealed what the conversation was about.
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