Lord Nicholas Monson said his son, Rupert Green, had suffered from psychosis as a result of an addiction to skunk - a strong form of cannabis.
Rupert was assessed in hospital and it was agreed he needed a bed, but staff could not find one, Lord Monson says.
Surrey and Borders NHS Trust said it was sorry and was investigating.
Rupert, who was a biology student at the University of Essex, died last month.
Lord Monson told BBC Radio 5 live's Emma Barnett that he had noticed a change in his son's behaviour last year when he had not heard from him for a while.
He said: "Suddenly I got a call from him and he apologised for not contacting me.
"He said he'd failed his biology exams at university.
"He said 'Spies are trying to break into my computer and I think spies are also in my computer'. Everything about him had changed."
Lord Monson says his son was assessed at hospital.
He said: "The nurse assessed Rupert and said that he really does need to go into a hospital and rang round all the hospitals and none of them would take him.
"They said they didn't have any beds... the following day he was assessed by his local psychiatry team and they decided if Rupert just carried on taking the pills, then the psychosis would subside."
Lord Monson said his son killed himself a few days later.
Jo Young, director of quality at Surrey and Borders Partnership NHS Foundation Trust, said: "I am sorry for what happened and we are undertaking an internal investigation into Rupert's care and treatment with us.
"This will help us understand where we can learn lessons to further improve our support of people such as Rupert."
Lord Monson has now lost two children in recent years after his eldest son, Alexander, died in police custody in Kenya in 2012 and said he is "very angry" with himself about what has happened.
An inquest last year found that Alexander, aged 28, had died "violently" at the hands of the police.
Since Rupert's death, Lord Monson has launched a campaign calling for skunk to be reclassified from a Class B drug to Class A.
He said: "In the old days, the psychoactive substance in cannabis was about 1.37% and now it's 22% with skunk... they have taken what is actually quite a decent plant and turned it into a monster."
Lord Monson also wants milder forms of cannabis to be legalised.
He said the reaction to his campaign has been "extraordinary" and he is shocked by the amount of families who have come forward with similar drugs problems.
Lord Monson added: "I know of three friends of mine... one whose son, because of skunk, tried to kill himself.
"His brave mother thought: 'I'm going to try this stuff [skunk] and see what it's like'. And she says it was an absolute nightmare.
"She said it was like confronting the devil himself when she looked in the mirror.
"There's another one who's got long-term psychiatric problems and will probably never be the same again.
"And there's another one who's been dabbling with it at college and he's been suffering from suicidal tendencies and his mother is trying to find a safe place away from the college."
A Home Office spokesman said: "This government has no plans to legalise cannabis.
"There is a substantial body of scientific and medical evidence to show that it is a harmful drug which can damage people's mental and physical health."
Details of organisations offering information and support with addiction or emotional distress are available at BBC Action Line, or you can call for free, at any time to hear recorded information for addiction on 0800 155 947 or emotional distress on 0800 066 066.
A hereditary peer has told the BBC his 21-year-old son killed himself days after being turned away from hospital because of a lack of beds.