The Disability Benefits Consortium, which represents more than 100 charities and organisations, said: "These immoral and devastating benefits cuts will push more disabled people into poverty, and worsen people's health."
The SNP said the measures would "harm the most vulnerable" and "mark the start of a new era of austerity cuts".
Debbie Abrahams, the Labour MP who chairs the Commons work and pensions committee, argued there were "more compassionate ways to balance the books rather than on the back of sick and disabled people".
However, other Labour MPs agree with the government that there is a moral case for reforming the welfare system to encourage people into work.
The Conservatives said the changes were "too little, too late" and needed to be "tougher".
Shadow work and pensions secretary Helen Whately asked why the government was only planning to save £5bn annually, when the total bill for health and disability benefits is forecast to rise to more than £100bn a year by 2029/30.
Liberal Democrat work and pensions spokesman Steve Darling said: "If the government was serious about cutting welfare spending it would get serious about fixing health and social care and the broken Department of Work and Pensions."
Some of the changes will require new legislation, raising the prospect of a rebellion by some Labour MPs during votes in Parliament.
However, the size of the government's majority limits the threat from a vote.