The government said any decision should await a report being written by the former bishop of Liverpool James Jones.
Andy Burnham, who called for the law, criticised the failure to include it in the Policing and Crime Bill.
He said it was wrong Margaret Aspinall had to use compensation for her son's death towards the "fight for justice".
The shadow home secretary said Ms Aspinall, whose son James died in the 1989 disaster, had to put the Â£1,000 she received in compensation towards legal fees and was then "living on the breadline" trying to cover the cost of fighting for justice.
The Leigh MP said the 27-year fight Hillsborough families had to endure showed how the odds were stacked against families seeking the truth.
It has been revealed South Yorkshire Police spent at least Â£2.1m on legal fees representing its suspended Chief Constable David Crompton during the inquests into the deaths of the 96 fans who were killed.
The Labour MP said: "This Bill was an opportunity to make this country fairer, to even up the scales and tip them in favour of ordinary families and away from the establishment.
"I fear we have failed to do that."
He said Ms Aspinall, who is chairwoman of the Hillsborough Family Support Group, watched the debate from the public gallery, and was "disappointed" by the government's refusal to accept equal legal funding in to the Bill.
The Policing and Crime Bill gives police stronger powers and hands responsibility for fire and rescue services to elected police and crime commissioners.
It creates so-called super complaints which will allow organisations like charities to bring a large-scale complaint against the police and strengthens protections for whistleblowers so identities are protected while investigations are conducted.
Home Secretary Theresa May said the Bill would allow police to go "further and faster" and "radically reform" the complaints and disciplinary systems.
However, Mr Burnham said it was a "mixed bag" that included improvements but also missed opportunities including the equal legal funding for victims as well as a second Leveson Inquiry into the relationship between police and the press.
The Bill will now pass to the House of Lords for further scrutiny.
In April, jurors ruled the 96 victims of Hillsborough were unlawfully killed.
The Rt Rev James Jones, the former Bishop of Liverpool, led the Hillsborough Independent Panel now acts as the home secretary's adviser on Hillsborough.

The Snowdon Partnership launched a consultation on Thursday, outlining the ways it wants to promote and safeguard the area.
It follows concerns about visitor numbers, litter and strains on rescue teams, facilities and the environment.
Snowdonia National Park Authority said it hoped to create the "best plan possible for the area".
According to the partnership, Snowdon is the third most visited attraction in Wales with 582,000 visitors and the busiest mountain in the UK.
John Harold, the director of The Snowdonia Society, said the mountain has seen "a dramatic increase in visitors to the area over the last few years".
"There are issues surrounding the pressures of use. We are looking at litter, at footpath erosion, the need for people to be aware of their responsibilities and the challenges they are setting themselves when they set out on the mountain, at sustainable transport," he said.
"We celebrate the mountain - people do value it and enjoy their time here - so we are not interested in turning people away.
"We are looking at the positives and putting the limited resources that the partner bodies have to get the best outcome."
The partnership also wants to build on the success of tourism, saying about Â£69m is directly spent by visitors in local communities.
Tracey Evans, chief executive of charity The Outdoor Partnership, said: "Outdoor pursuit tourism is the biggest income generator in the area and it's about managing it properly, identifying those honey pot areas and dealing with the challenges.
"There is no one single answer. Do we need to look at an improved sherpa [bus] service, do we need a tourism tax where hoteliers and caterers come on board and the money is reinvested into the area?
"This partnership approach should go some way to addressing these issues."
Helen Pye, Snowdonia National Park Authority's partnerships manager, said: "People's views and ideas are extremely important to us which is why we're providing this opportunity for people to give their feedback on the draft plan.
"Ensuring input from others is so important so that we can produce the best plan possible for the area, and the plan as a result will evolve and be adapted as appropriate."
The online consultation runs until 7 July.
A decision on a "Hillsborough Law" to ensure legal funding for bereaved families at inquests where police are involved has been put on hold.
[NEXT_CONCEPT]
A drive is under way to protect the future of Snowdon following a rise in visitors to the mountain.