Combat Stress said its income has fallen from £16 million to £10 million in this financial year partly due to a cut in its NHS funding support, and said the decision to turn down new cases had been taken "with great sadness".
The charity said until 2018 it got more than £3 million a year from NHS England, but that now 90% of its funding comes from public donations.
The JPIMedia Investigations Team's investigation into military suicides led to national calls for the Government to do more to support current and former members of the forces suffering from mental health problems.
Combat Stress still gets more than £1 million from NHS Scotland and will continue to take on new cases there and in Northern Ireland.
It had been receiving around 2,000 referrals for treatment a year.
It will now send all new referrals from England and Wales to the NHS, which Combat Stress said "needs to demonstrate" it can deal with the increased demands.
"I don't believe the NHS can pick this up. That is why we exist," Sue Freeth, chief executive of Combat Stress, told the BBC.
She said that 80% of veterans who come to the charity have either used the NHS and have not had their needs met, or have felt unable to use NHS services.
Veterans' minister Johnny Mercer said he will hold an "urgent meeting" over Combat Stress's problems.
Last September, Mr Mercer, who is a former Army captain, vowed to provide veterans with the "best mental health care in the world" after saying post-combat stress "ripped apart" those he served with.
"I have an intimate understanding of the issues. I am not going to pretend for a minute that I have all the answers - I don't think anybody has got all the answers.
"But I am determined to get mental health care - whether it's in the veterans' community or the military community or indeed the NHS - to a place where we can offer those who have served in this country the best mental health care in the world."
Several organisations and charities have warned of a rise in the number of veterans taking their own lives.
A spokesman for the NHS said: "Our number one priority is providing the best care for veterans and, after listening to what they wanted and a competitive process, the NHS has rolled out new specialist services to every part of the country which have seen over 10,000 people to date and are funded by more than £10 million every year.
"For anyone who has served in the Armed Forces and may be experiencing mental health difficulties help is available through speaking to their GP or contacting the dedicated NHS services directly."