In his resignation letter to Prime Minister Theresa May, Alan Milburn, a former Labour Health Secretary, who took up the role in July 2012, said individual ministers such as Education Secretary Justine Greening have shown a deep commitment to social mobility, but added that it has "become obvious that the government as a whole is unable to commit the same level of support".
"[The government] is understandably focused on Brexit and does not seem to have the necessary bandwidth to ensure that the rhetoric of healing social division is matched with the reality," the letter said.
"I do not doubt your personal belief in social justice, but I see little evidence of that being translated into meaningful action.
"The need for political leadership in this area has never been more pressing. Social mobility is one of the biggest challenges facing our country today. It is not just the poorest in society who are losing out.
"Whole communities and parts of Britain are being left behind economically and hollowed out socially. The growing sense that we have become an ‘us and them' society is deeply corrosive of our cohesion as a nation."
The other board members standing down are former Conservative Education Secretary Baroness Shephard, Paul Gregg, a professor of economic and social policy at the University of Bath, and David Johnston, chief executive of the Social Mobility Foundation charity.
The government has said it had already decided to appoint a new chair next year and defended its record on social mobility.
Milburn has been a vocal critic of government policy in the past.
In 2014, the commission warned that there was "no way" the government could meet its own target to eradicate child poverty by 2020.
Last week, the commission's annual report Social mobility in Great Britain: Fifth State of the Nation Report found that London and its surrounding areas pulling away from the rest of the country in terms of children's life chances, while many other parts of the country are being left behind economically.
Milburn called on government to increase its proportion of spending on those parts of the country that most need it.
The government, under former Prime Minister David Cameron, had previously made plans to boost social mobility, through the so-called "life chances strategy". But, last December it emerged that the government had shelved the idea.
The strategy had been due to feature measures designed to address child poverty, including a plan to significantly expand parenting provision, and potentially introduce a voucher scheme for parenting classes. The government had also said it would contain future policy on children's centres, a consultation on which had been due to launch in autumn 2015.
At the time, the government said elements of the life chances strategy would instead feature in a social green paper, although this has yet to materialise.
Milburn said he plans to establish a new social mobility institute, independent of the government and political parties, to work on the issue.