Erdogan, who blames the plot on the US-based preacher Fethullah Gulen, also described the coup as a "scenario written from outside" in an allusion to foreign involvement.
Turkey meanwhile issued arrest warrants for about 100 staff, including doctors, at Ankara's main military hospital, in a new phase of the crackdown after the failed coup that has seen some 18,000 detained and caused international consternation.
Erdogan lashed out at Germany's judicial authorities for not allowing him to address via video conference a weekend rally in Cologne in his support.
He accused Germany of allowing leaders from the outlawed Kurdistan Workers Party (PKK) to address previous events via video conference.
He said Turkey had previously also handed Germany a list of more than 4,000 wanted militants without having any response.
The president, who had previously alluded to foreign states' involvement, gave his strongest indication yet that external powers could have been played a role.
Turning directly on Washington, he asked: "How can it be, when we are strategic partners and I ask you on behalf of my country to hand someone over on the basis of a national security strategy document, you keep on hiding and sheltering him?"
Already strained ties between NATO allies Turkey and the United States have been aggravated by the failed putsch, with some government ministers even alleging Washington could have had a hand in the plot, which US officials firmly reject.
Police searched the Gulhane Military Medical Academy (GATA) hospital in the capital and detained some 50 staff including military doctors, the state-run Anadolu news agency reported, quoting police sources.
A Turkish official, speaking on condition of anonymity, confirmed detention warrants had been issued for 98 staff.
The official said staff there were suspected of helping fast-track Gulen supporters into the military by giving them favourable medical reports.
Similar claims have also been made about military schools where officials say exams were rigged. Almost half of Turkey's generals were fired in the wake of the coup.
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