Microsoft was awarded the Joint Enterprise Defense Infrastructure (Jedi) contract in October.
A month later, Amazon filed a notice in the US Court of Federal Claims, claiming the process had contained deficiencies and “unmistakable bias”.
It accused President Trump of political interference.
Amazon had been the favourite to win the vast contract, which is worth $10bn (£7.6bn) over the next 10 years. It now wants the deal put on hold until the court rules on its protest.
In a statement, Amazon’s cloud division Amazon Web Services said: “It is common practice to stay contract performance while a protest is pending, and it’s important that the numerous evaluation errors and blatant political interference that impacted the Jedi award decision be reviewed.
“AWS is absolutely committed to supporting the Department of Defense’s modernisation efforts and to an expeditious legal process that resolves this matter as quickly as possible.”
Microsoft declined to comment.
In July, President Trump told reporters that he was “getting tremendous complaints” about the possible deal between the Pentagon and Amazon.
Previously he had been critical of Amazon and its founder Jeff Bezos – who owns the Washington Post.
In a statement at the time the contract was awarded, the Pentagon said that all offers “were treated fairly”.
Defense Secretary Mark Esper has also rejected the accusations of bias, saying the Pentagon made its choice without external influence.
Investment bank Wedbush said it did not think its case will change the decision.
“While Amazon will continue to fight this issue in ‘Jedi-gate’ and possibly drag out the inevitable start of Jedi, we ultimately believe this is a paradigm changer for Microsoft.”
What is Jedi?
The Department of Defense wants to replace its ageing computer networks with a single cloud system.
Under the contract, Microsoft will provide artificial intelligence-based analysis and host classified military secrets among other services.
It is hoped that Jedi will give the military better access to data and the cloud from battlefields.