Tom Evans could be seen sleeping behind Alfie in Alder Hey Hospital.
Tom Evans said his 23-month-old son survived for at least nine hours with no assistance, and that doctors are providing oxygen and hydration.
He said: 'We got rejected yesterday to go to Italy unfortunately.
According to the BBC, Alfie's parents will meet with doctors later today to discuss taking him home.
Doctors at Alder Hey have argued that Alfie can not recover from his "relentless and progressive" brain condition that has left him in a semi-vegetative state.
Alfie doesn't need intensive care no more'. Some people say it's a miracle, it's not a miracle, it's a misdiagnosis.
"We can not let a helpless child die right in front of the world's eyes with no one willing to help", she added.
The family now have the support of hundreds of protestors who have gathered outside Alder Hey in support of having Alfie airlifted to Italy, despite British doctors disagreeing with the transfer.
Evans slammed the United Kingdom court system and Alder Hey Hospital for what he considers their heartlessness.
The little boy, who is nearly two-years-old, has been legally prevented from travelling overseas for medical treatment after life support was withdrawn at Alder Hey Hospital on Monday.
Mr Evans said the family did have "appeals to explore".
I am not alone in believing that when British law can displace the rights of parents then British law must be changed.
A lawyer for Alfie Evans' mother Kate James told three Court of Appeal judges that James hoped the courts would "invite the hospital to take the appropriate steps".
So far, the similarities to the Charlie Gard ordeal in London previous year have been glaring: Charlie suffered brain damage; his doctors withheld treatment; his parents objected; two hospitals, including Bambino Gesù, offered to take him. For months, they have reiterated that further medical treatment for Alfie would not be in his "best interest".
In the caption to the post, she lovingly wrote: "How gorgeous is this boy tho (sic)".
Last night a spokesperson for the family said Tom had "fallen asleep next to his little boy" and would give a statement on Thursday morning.
Alfie has been in Alder Hey with a rare undiagnosed degenerative neurological condition since December 2016. He also said an air ambulance was on standby "at the request of the Pope".
"We are very grateful and we appreciate all the support we have received from around the world, including from our Italian and Polish supporters, who have dedicated their time and support to our incredible fight".
Lady Justice King said there was "acceptance" that Alfie was dying.