Naz Shah has apologised for sharing the graphic showing Israel's outline superimposed onto a map of the US with the comment "problem solved".
It was shared in 2014, before she became Bradford West MP.
The Board of Deputies said the post, highlighted by the Guido Fawkes political website, was "appalling".
The graphic said relocating Israel would be a "solution to the Israel-Palestine conflict", and that it would allow Palestinians to "get their life and their land back".
In a statement, Ms Shah said: "This post from two years ago was made before I was an MP, does not reflect my views and I apologise for any offence it has caused."
Labour confirmed she had stepped down as Parliamentary Private Secretary - an unpaid backbench assistant - to shadow chancellor John McDonnell.
In a second statement, the MP said: "I made these posts at the height of the Gaza conflict in 2014, when emotions were running high around the Middle East conflict.
"But that is no excuse for the offence I have given, for which I unreservedly apologise."
She said she would be expanding her "existing engagement and dialogue with Jewish community organisations and will be stepping up my efforts to combat all forms of racism, including anti-Semitism".
The Board of Deputies of British Jews said the comments were "simply appalling", calling for an urgent meeting "for clarification of her views on Israel and the UK Jewish community".

In Burnell's first action of the season, he beat a field of more than 80 swimmers at Emirates Palace Bay.
The 23-year-old, who represented Team GB at Rio 2016, beat Dutchman Ferry Weertman in a sprint to the finish.
"At the end of the day, it's about ticking that box and getting the first one under my belt since Rio," said Burnell.
"I am out there to win every single race I go into, I am not out there to come second, third or fourth.
"So for me it's about turning up to races in the best possible shape I can and showing people what I can do."

The oil firm said revenues in the year fell 27% to £1.6bn as the price of oil continued to fall in 2015.
It is the second year in a row the oil company has reported a loss. In 2014, it reported a loss of £2bn.
Tullow also warned there was a risk it could fail to comply with its financial covenants this year.
Financial covenants are agreements that companies make with their lenders to maintain a stable financial position.
Oil prices have slumped by 70% since the middle of 2014, with many of the oil majors cutting back investment on exploration and production.
That has also led to the loss of more than 5,000 jobs in the North Sea oil industry.
Tullow said it could cut annual capital expenditure to as low as $300m in 2017, down from $1.1bn planned for 2016, if market conditions did not improve.
The firm said it had reduced headcount of 37% and was on track to deliver cash savings of around $500m over three years.
It also said it would pay no dividend in 2015.
Tullow Oil chief executive Aidan Heavey said: "Our challenge in 2016 is to be equally robust in responding to the uncertainties that remain in the sector."
He added: ". As we look ahead, we have a portfolio of world class, low cost oil assets which will produce around 100,000 barrels per day in 2017 and  a major position in one of the world's newest, low cost, oil provinces in East Africa, both enabling us to create substantial value."

15 January 2016 Last updated at 07:44 GMT
He described walking in Space as "absolutely fantastic" and he was "giddy with excitement."
Luca was selected as an ESA astronaut in May 2009 and travelled to the International Space Station in 2013
On his mission he took part in two spacewalks, orbited Earth 2,656 times and travelled more than 110 million km in total.
A Labour MP has stepped down as an aide to the shadow chancellor over a Facebook post suggesting Israel should be moved to the United States.
[NEXT_CONCEPT]
Britain's Jack Burnell won gold at the Abu Dhabi leg of the 10km Marathon Swimming World Cup.
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Oil and gas exploration firm Tullow Oil has reported a pre-tax loss of £1.3bn for the year to 31 December as low oil prices bit into revenues.
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The European space Agency astronaut, Luca Parmitano, spoke to Newsround about his spacewalk experiences.