Robert Buckland QC apologised for the failings this week, with his words since described by opposition MP David Lammy as 'meaningless', calling for him to resign if the rates do not improve.
Currently, just 1.6 per cent of rapes and attempted rapes reported to police result in a charge.
The secretary of state has since hit back at calls for his resignation, saying Mr Lammy was playing "low politics" over the issue.
Earlier this week, ministers set out plans for a “system and culture change” after convictions for rape and lesser offences in England and Wales hit a record low.
Speaking to BBC's Laura Kuenssberg, Mr Buckland admitted the current problem was "not good enough" and that major changes - both cultural and financial - were needed.
“The first thing I need to say is sorry,” the QC said.
“It’s not good enough. We’ve got to do a lot better. And the plan that we are outlining in that review is all about taking practical steps to change the situation and to address the causes for this failure.”
Following Mr Buckland's comments, Labour MP David Lammy said: “The Justice Secretary’s crocodile tears will mean nothing if the Government fails to reverse its disastrous failure of rape victims.
“The Conservatives’ decade of cuts to the justice system has let rapists and other violent criminals off the hook while denying victims justice.
“Rape convictions and prosecutions have more than halved in three years. If he cannot reverse these figures within a year of his apology, the Justice Secretary should do the honourable thing and resign.”
Mr Buckland struck back on Sunday, refusing to say whether he would resign if he fails to meet a target to rectify the issue, and describing the call as “constitutionally illiterate”.
“Decisions made to investigate and prosecute are made by the independent police and their operational work and the CPS, which is independent,” he told Sky’s Trevor Phillips On Sunday.
“If there was any suggestion that prosecutions were being brought about because of political pressure on me frankly that would make convictions unsafe – it’s a ridiculous argument.”
Pressed again if he would resign if he did not meet a target, Mr Buckland said: “The idea that somehow a resignation or political pressure should be brought to bear on independent prosecutorial decisions is not only bad politics but it’s actually dangerous. I’m not going to engage in that level of debate.”
The latest CPS figures for 2019-20 show 1,439 suspects were convicted of rape or lesser offences in England and Wales last year – the lowest level since records began.
That figure was down from 1,925 the previous year, despite reports of adult rape to police almost doubling since 2015-16.
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