The Home Secretary, who is also the equalities minister, said: "We are scrapping the socio-economic duty for good. It would have been just another bureaucratic box to be ticked."
The measure, introduced by Labour's equalities minister Harriet Harman, was intended to force public authorities to reduce inequality by taking into account disadvantage and poverty when making decisions about policies.
The clause would also have applied to job centres, where officers would have had to assess whether an unemployed person would be worse off if their benefits were cut as a result of the government's new welfare policies.
By scrapping the clause, Ms May has removed the possibility of properly monitoring the impact of the government's welfare overhaul on the poorest.
Disability Alliance director of policy Neil Coyle warned that Ms May risked plunging many more vulnerable people into poverty.
He said: "We believe that certain aspects of the welfare changes will risk impoverishing thousands of people.
"The socio-economic duty would have monitored and prevented the implementation of any proposals that would have led to more poverty among disadvantaged people."
Child Poverty Action Group chief executive Alison Garnham said: "Far from being a box-ticking exercise, the duty would have given power to the powerless, making sure the voices of the most disadvantaged have a say in how government makes decisions.
"The message now is that government and public bodies can ignore the powerless and dispossessed and the bureaucrats can carry on regardless."
GMB's equalities officer Kamaljeet Jandu warned Ms May's announcement was "a dangerous sign of things to come."
Unison general secretary Dave Prentis added: "This watering down of the Equality Act is yet another example of the Tories hitting the poorest hardest."
Campaigners are also concerned that the government may scrap another part of the Act which requires private-sector employers to disclose whether they pay women as much as men.
The government's failure to implement the Act in full "undermines every speech coalition ministers ever gave endorsing the notion of a fairer Britain," women's group The Fawcett Society said.