The handwritten note was left in their dressing room following a goalless draw in their second Group G game.
Iran are competing at the World Cup while their country and the United States - who are co-hosts along with Canada and Mexico - are in negotiations to end their war.
"From the ancient Persia of thousands of years ago to the civilised Iran of today, the spirit of Iran remains alive and steadfast," read the message, which was released by the Football Federation of Iran (FFIRI).
"We came to Los Angeles with pride, competed with honour, and leave with dignity," the note continued.
"Thank you Los Angeles for your hospitality.
"And thanks you to every Iranian who gave their heart, voice and soul for Iran throughout these 180 minutes.
"May peace, respect and friendship prevail among all nations."
The message also made reference to an attack on a girls' school in Minab, where at least 168 people were killed on the first day of the war.
Iran played their opening match - a 2-2 draw with New Zealand - at Los Angeles Stadium, and will play their third and final group game against Egypt in Seattle on June 27.
Their involvement in the World Cup has been plagued with uncertainty, linked to the war in the Middle East and related security concerns.
Multiple "integral" members of their backroom staff were denied entry visas for the US while the FFIRI called on Fifa to "uphold the principles of neutrality, fairness, and established regulations" after its allocation of tickets was revoked on the eve of the tournament.
Iran subsequently switched their base for the World Cup from Arizona in the US to Tijuana in Mexico, and face strict travel restrictions during their campaign.
They are allowed to fly into the US a day before they have a game and have to leave the country on the same day as the match, under the conditions of their visas.
This led Iran to head coach Amir Ghalenoei saying they are the "most oppressed" team at the tournament following the draw with New Zealand.
However, Andrew Giuliani, the executive director of the White House's World Cup taskforce, says there are "discussions going on" about the timings for Iran's travel to and from their game against Egypt.