Aside from attacking the Washington-based publication directly, Trump harshly criticized Amazon during his campaign, saying the e-commerce giant operated a monopoly with an unfair tax shelter that's somehow propped up by Bezos' ownership of the Post.
During one campaign rally in February 2016, Trump told the crowd: "If I become president, oh [does Amazon] have problems. They're going to have such problems." He added that Bezos only bought the Post to have "political influence."
Trump is now calling out Amazon for avoiding so-called internet taxes, what was once a much larger and controversial issue for the e-retailer.
A representative from Amazon did not immediately respond to a request for comment.
However, as pressure built in individual state houses, Congress and courts to allow states to require retailers to collect the tax on out-of-state sales and pass it on to state governments, Amazon has accelerated changes in its tax policy.
Starting April 1 this year, Amazon began collecting sales tax nationwide, calming much of the prior controversy. So it remains unclear what Trump was referencing when he accused Amazon on Wednesday morning of not paying "internet taxes."
In general, online retailers fall under two different tax systems today — retailers located out of state pay no sales tax, while those with some type of link to the state must pay it. Amazon having distribution centers in certain states, but not others, had put the company in a tricky spot.
"This is a personal thing. ... He's going after Jeff," Gene Munster, co-founder and managing partner of Loup Ventures, told CNBC's "Squawk on the Street." "It's all about personal vendetta against The Washington Post."
But Munster added that Amazon shouldn't be impacted by this in the slightest way. "Bezos is just going to double down ... [he] doesn't have anything to worry about."
Everyone knew the increase in sales taxes was coming, for example, so this shouldn't impact unit growth for Amazon at all, Munster said.
Meanwhile, Bezos hasn't been shy about his anti-Trump views either. At one point during the election, Bezos tweeted that he'd save a seat for Trump on his Blue Origin spacecraft, with the hashtag "sendDonaldtospace." At another time, he called out Trump's attempts to intimidate the press.
Just last week, Bezos, among other tech CEOs, attended a meeting with the president in the White House, where executives discussed key issues impacting Silicon Valley, like tax reform.
Amazon shares inched down less than 1 percent after Trump's tweet Wednesday morning.
— CNBC's Ari Levy contributed to this report.
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