Al Rumaithi, who was elected on June 8, said his resignation would allow fresh blood to be brought in to inspire new ideas to build a strong national team.
"My resignation stems from my personal conviction, without influence or pressure," he said in a statement. "To tell you a secret, I have already submitted my resignation more than once.
"I have submitted my resignation as recently as in September, the first time after the defeat to Lebanon and thereafter following the team's exit from the 2014 World Cup qualifier."
The UAE's hopes of qualifying for Brazil 2014 were ended by their home defeat to Kuwait on September 2 and their loss in Lebanon four days later.
They are now without a win after five games, following two losses to South Korea and another to Kuwait, with one game remaining, against Lebanon on February 25.
Al Rumaithi said he was confident the FA could find someone "more competent to carry on and live up to our ambitions and the ambitions of the UAE fans".
He said: "I found it wiser to hand over the reins to someone else after having a strong feeling that I'm unable to achieve such ambitions.
"It's not part of my beliefs to insist on remaining in this position which I consider as a responsibility and not an honour.
"I undertook in front you to achieve all our targets. And therefore, I have to be honest with myself above all and with you that my stand is final and unchangeable."