Dimitry Rogozin warned that Russian tech and expertise are needed to prevent the ISS from crashing down onto the US or Europe.
Rogozin is director general of Roscosmos, the Russian equivalent of the National Aeronautics and Space Administration of the US.
"If you block cooperation with us, who will save the ISS from an uncontrolled deorbit and fall into the United States and Europe?" Rogozin tweeted on Friday, following US President Joe Biden's announcement of sanctions over Russia's invasion of Ukraine.
Musk weighed in by replying to Rogozin's threat with the SpaceX logo, according to the news portal of the US Sun.
When a Twitter user asked the billionaire if this meant SpaceX would keep the ISS from falling onto Earth, Musk replied: "Yes."
He also replied to a tweet that showed the ISS without the Russian segment and a SpaceX Dragon attached instead.
"Good thread," Musk wrote.
Russia is a key part of the 15-nation partnership that has kept the ISS orbiting Earth for 23 years but relations are at an all-time low because of Russia's decision to invade Ukraine.
On Thursday, Biden unveiled new sanctions that he said would "degrade" Russia's "aerospace industry, including their space program".
Rogozin then warned that Russian engines control the station's orbit and location.
"There is also the possibility of a 500-ton structure falling on India and China," he added.
"Do you want to threaten them with such a prospect? The ISS does not fly over Russia, therefore all the risks are yours. Are you ready for them?"
The comments are an escalation from remarks made by the Roscosmos boss just days before, when he tweeted that Russia "treasures" its relationship with NASA and would "make every effort to continue as before" regardless of "disagreements between our countries".
On Thursday, UK Prime Minister Boris Johnson was also pressed on the issue, telling MPs: "Hitherto I have been broadly in favor of continuing artistic and scientific collaboration, but in the current circumstances it’s hard to see how even those can continue as normal."
However, NASA has sought to ease tensions, saying sanctions won't affect ISS operations.
"NASA continues working with all our international partners, including the State Space Corporation Roscosmos, for the ongoing safe operations of the International Space Station," a rep for the agency explained.
"The new export control measures will continue to allow US-Russia civil space cooperation. No changes are planned to the agency’s support for ongoing in-orbit and ground station operations."
In recent years, NASA has been trying to reduce its reliance on Russian rockets to send its astronauts into space, instead getting Elon Musk's SpaceX to build them.
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