In the beginning of Shadow the Hedgehog, the president is contacted by his Secretary, who reports on alien invaders getting pushing back from the Westopolis and reporting alien invasions in five other cities in the country. The president tells her to command the generals never to surrender to terrorists.
Depending on, if the player chooses to follow True Dark Story in the game, Shadow and Black Arms manage to take over the Space Colony ARK and take over Earth, beginning by defeating all leaders who resist their invasion. During this, Black Arms activate the Eclipse Cannon and fire directly to the White House, destroying Central City along with it. However the President has got time to escape from the White House before this to the GUN Fortress, where Commander tells him that the citizens of Central City have evacuated the day before to the safe zone.
If the player chooses Semi-Dark story, the White House does not get destroyed, but the President will get evacuated from the building by GUN Commander and GUN Soldiers to the Air Fleet due to Central City being lost to the Black Arms. Outside on the garden of White House, Black Doom despises their escape and commands Shadow to follow them to the Air Fleet and destroy the President's escape pod.
In the anime series Sonic X and its comic series published by Archie Comics, the White House is a location on Earth and the residence of the President. In this media, the White House is also completely identical to the real-life building. During Dr. Eggman's time on Earth, the White House host's numerous crisis reports related to Eggman.
In Shadow the Hedgehog, the President is seen to have photo on his workbench, showing Sonic and Shadow posing near of White House, while the President remembers their feat moments from Sonic Adventure 2. Though there hasn't been any given sign of when this photo was taken, it stands to reason that it was taken sometime after Sonic Heroes, as Shadow reappeared in previously mentioned game after his supposed death from the feat that made him the hero the President acknowledged.
The cinematic cutscene showing the destruction of the White House from Shadow the Hedgehog has been directly influenced by 1996 science fiction film Independence Day. According to Takashi Iizuka, the game gains numerous film references.
Because the game identifies Shadow the Hedgehog's final story as part of the series' continuity, it is unclear if the building was destroyed or not, as it has made no appearances in later installments.
If one looks very closely at the flags in the White House picture, one can make out their design. This design is most likely the real flag of the United Federation, rather than Empire City's flag, unless this has been reteconed. Either way, this flag appears as the United Federation flag in the Archie Comics.