Sahin Alpay and Mehmet Altan had been accused, in separate cases, of links to US-based Muslim preacher Fethullah Gulen who Ankara says masterminded the failed 2016 coup attempt. Gulen denies the charges.
Rights campaigners hoped the Constitutional court ruling could set a major legal precedent and help could help dozens of other journalists arrested in the crackdown.
But two Istanbul criminal courts late yesterday defied the ruling and ordered that both writers should for now remain behind bars.
"With this decision the Constitutional Court has crossed over its legislative and constitutional limits," Bozdag wrote on Twitter. "This court should not act as an appeals court."
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He described the decision as "bad and wrong", saying it was reminiscent of the same court's ruling in February 2016 to order the release from jail of the former editor of the Cumhuriyet daily Can Dundar.
President Recep Tayyip Erdogan had at the time fumed over that decision, saying he had "no respect for it".
Most of its 17 members were appointed during the rule of Erdogan's predecessor Abdullah Gul or the previous president Ahmet Necdet Sezer.
"Now we know why they weren't released last night," commented Emma Sinclair Webb, the Turkey director of Human Rights Watch, on Twitter after Bozdag's remarks.
The Istanbul courts said that the men could not be released as Constitutional Court had not formally communicated its rationale.
But the top court hit back with a rare message on Twitter, saying its full ruling could be read by anyone on the internet.
Alpay, 73, is a political scientist who wrote a column for the now closed staunchly pro-Gulen Zaman newspaper.
Mehmet Altan, 64, has written books on Turkish politics. His brother Ahmet, a novelist and journalist for some of Turkey's leading dailies, is also being held in the same case.