Chris Gard and Connie Yates, both in their 30s from Bedfont, west London, wanted 10-month-old Charlie, who suffers from a rare genetic condition and has brain damage, to undergo a therapy trial in the US.
But specialists at Great Ormond Street Hospital (GOSH) said the therapy was experimental and would not help.
Charlie's parents had asked European court judges in Strasbourg, France, to consider their claim after judges in the High Court, Court of Appeal and Supreme Court in London ruled in favour of GOSH doctors.
But on Tuesday the European Court of Human Rights refused to intervene.
Ms Yates described the day Charlie was born, August 4 last year, as the best day of their lives but said June 30 2017, would be the worst.
The couple say they have been denied their final wish to be able to take their son home to die.
Alongside a video posted on YouTube on Thursday, Charlie's parents wrote: "We are utterly heartbroken spending our last precious hours with our baby boy.
"We're not allowed to choose if our son lives and we're not allowed to choose when or where Charlie dies.
"We, and most importantly Charlie, have been massively let down throughout this whole process."
Charlie's plight has touched many people and the family received donations totalling more than £1.3 million to take him to the US for therapy.
Campaigners also pledged their support to the family on social media with hashtags including #charliesfight #Fight4Charlie and #LetCharlieGoHome.
His parents said Charlie would die "knowing that he was loved by thousands", adding "thank you to everyone for all your support!"
In the video the couple said they felt they were being rushed, despite having been promised they would have all the time they needed to say goodbye to their son.
A hospital spokeswoman said: "As with all of our patients, we are not able to and nor will we discuss these specific details of care.
"This is a very distressing situation for Charlie's parents and all the staff involved and our focus remains with them."