The body faced disapproval for not lighting up the arch in the colours of the Israeli flag for England's friendly with Australia.
Bullingham acknowledged the call had caused hurt to the Jewish community.
A period of silence was instead held.
"This was one of the hardest decisions we've had to make," Bullingham said.
"The last thing we ever wanted to do in this situation was to add to the hurt.
"This week has made us question whether we should light the arch and when, and we'll be reviewing that in the coming weeks."
More than 1,400 people were killed in Israel in attacks launched by militant group Hamas last week, while the health ministry in Gaza says more than 3,700 people have been killed there since.
In the past, the FA has llit up the arch to commemorate events including Remembrance Day, International Women's Day or to pay tribute to victims of the war in Ukraine or of terror attacks such as in Paris in 2015.A message of peace and unity was displayed on Wembley's big screens before kick-off while both sides adorned black armbands
However, Bullingham, speaking at a Leaders Week conference at Twickenham Stadium, said the FA had sought "expert guidance" on the Israel-Gaza issue before reaching its decision.
"We had a long board meeting on the Wednesday night and heard from experts on what is one of the most complicated geopolitical conflicts on Earth," he added.
"It's worth noting that the Australians had upcoming games against both Palestine and Lebanon, so their desire for neutrality was obviously incredibly strong.
"We all felt then, and we all feel now, that football should stand for peace and humanity and the wish to show compassion for all innocent victims of this terrible conflict.
"I recognise our decision caused hurt to the Jewish community who felt that we should have lit the arch, and that we should have shown stronger support for them.
"We aren't asking for everyone to agree with our decision, but to understand how we reached it."
Lucy Frazer, the Cabinet minister responsible for sport, criticised the FA's decision, while Rabbi Alex Goldberg resigned from an FA faith in football group over its response.
But Bullingham questioned why the spotlight shone so strongly on football, and not on other sports, in such sensitive and potentially divisive situations.
"It would be easy for football to ask why we're the only sport being talked about in this way, particularly when rugby and cricket are in the middle of their World Cups," he said.
"However, you have to understand, and we understand, that the power of football means it will always be in the spotlight. And that's just something we have to accept."
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