Mr Jones has accused the platforms of unfair censorship of his accounts.
The radio host is best known for spreading unsubstantiated allegations about tragic events, including 9/11.
He is currently being sued for defamation by three parents whose children were killed in the Sandy Hook Elementary school attack, which he has repeatedly claimed was a “giant hoax”.
In a series of tweets on Tuesday, Twitter CEO and co-founder Jack Dorsey explained the platform’s decision, confirming the platform would not be following in the footsteps of others like Apple and Spotify and removing Mr Jones’ and InfoWars’ content.
Mr Dorsey said the accounts had not violated the platform’s rules, but vowed to suspend them if they ever were in contravention.
In his explanation, Mr Dorsey said it would be wrong to “succumb and simply react to outside pressure” rather than stick to the company’s codified principles.
He also implied one-off actions risked fuelling new conspiracy theories in the long run, and said it was critical for journalists to “document, validate and refute” unsubstantiated rumours like the ones spread by Mr Jones “so people can form their own opinions”.
Some users replied angrily to the CEO’s tweets, accusing the platform of ignoring hate speech and labelling his explanation a “cop out”.
Several right-wing commentators have suggested tech firms’ ability to block a publisher pose a challenge to the free speech laws in the US.
This is not the first time Twitter has faced similar controversy.
Last year it suspended its verification system after being accused of amplifying the content of far-right accounts with official “blue ticks”.
In Tuesday’s tweets Mr Dorsey admitted the platform had been “terrible at explaining our decisions in the past” but said it was committed to fixing itself.
A number of well-known far-right figures including Tommy Robinson have been banned from Twitter in recent months as part of a wider crackdown on hateful and abusive content on the platform.
Earlier this year, the CEO asked for the public’s help with tackling online abuse.