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The phrase "back up the assertion" is correct and usable in written English.
It can be used when you want to provide evidence or support for a claim or statement made in a discussion or argument. Example: "To strengthen your argument, you should back up the assertion with relevant data and examples."
Exact(16)
But the senator has provided no evidence to back up the assertion.
The military said armed men on motorcycles had started the assault, but it offered no evidence to back up the assertion.
The government will likely need actual witnesses, people who worked at SAC and are willing to back up the assertion that the firm encouraged them to break the law.
None of the text he cites provides any data to back up the assertion that "the federal and state governments are either the main employers or at least the chief sources of income — whether through emergency rooms, rural health clinics, public schools, or social-service offices".
But the FCC has not produced detailed logs or other analyses to back up the assertion, and the agency's own commissioners have claimed the irregularities could have allowed fake messages sent by automated bots to crowd out feedback from American citizens.
So far, however, officials have not produced transcripts or recordings of the conversations to back up the assertion, according to witnesses who have been interviewed.
Similar(44)
But in each case, the officials have offered no details to back up the assertions.
Nor are there any links to back up the assertions in the text of all these achievements.
They were presented without any data or information to back up the assertions, and reveal some of the materials the Trump administration reviewed as it was crafting its trade policy.
Last week, several senior administration officials made statements suggesting there was new proof of links between Al Qaeda and Baghdad, but did not offer any details to back up the assertions.
Neither paper reported any evidence to back up the assertions of unnamed officials, who are simply to be believed.
More suggestions(1)
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com