Edwards, the Labour MP for Rochester and Strood, said: “Parliament has dropped the ball today.
“It will be picked up again as it must be, because the law as it stands is simply too cruel and too unjust to be allowed to stand.”
Those in favour of a change in the law have said they would try to bring forward legislation again, with Kim Leadbeater, the Labour MP who introduced the original bill, telling the another attempt could come as soon as next year.
Labour MPs Ashley Dalton, Dame Meg Hillier and Jess Asato, who all opposed what they described the “unsafe and unworkable” bill, welcomed MPs’ decision.
In a joint statement, they said: “This is not a moment for celebration, we understand the deep concerns of so many people across the country who have campaigned hard for a change in the law.
“But the truth is none of the relevant royal medical colleges, professional groups or experts will attest to the safety or workability of this bill.”
They said there is a need to “fix our NHS and sort out social and palliative care before we even consider introducing assisted dying”.
Sarah Wootton, chief executive of Dignity in Dying, said “thousands of terminally ill people will continue to suffer against their wishes in their last days and weeks”.
She added she still believes “law change is inevitable”.
Care Not Killing chief executive Gordon Macdonald described the vote as “decisive”.
He said: “Our hope is that MPs now focus on repairing palliative care in the UK and securing the necessary funding to make sure that no matter what part of the country you live in, you can access the best possible care available and are treated with dignity and respect.”