The International Association of Athletics Federation (IAAF) decided not to lift the suspension, imposed after accusations of state-sponsored doping.
Individual athletes can compete as neutrals if they prove they are clean.
The Russian president called on other bodies, including the International Olympic Committee (IOC), to intervene.
The IOC executive board said it would hold a telephone conference on Saturday to discuss the issue ahead of a full IOC summit in Lausanne on Tuesday.
"There are universally recognised principles of law and one of them is that the responsibility should be always personified," said President Putin.
"The people who have nothing to do with violations, why should they suffer for those who committed the violations?
"I'm assuming that we'll have a discussion with our colleagues in the World Anti-Doping structure and I hope for a suitable reaction from the International Olympic Committee."
The country was suspended by the IAAF in November 2015, after an independent World Anti-Doping Association (Wada) report depicted a culture of widespread doping, with even the secret services involved.
After that, Russia introduced reforms including an overhaul of the rules, the introduction of independent testing, and anti-doping lessons in schools.
A task force has been studying those reforms but a fresh Wada report, issued on Wednesday, made more damaging claims.
Although significant progress has been made to meet the IAAF's criteria, it said, work still remains. In particular:
- The deep-seated culture of tolerance for doping appears not to have changed. The head coach of the athletics team and athlete appear unwilling to acknowledge the extent of the doping problem.
- A strong and effective anti-doping infrastructure capable of detecting and deterring doping has still not been created.
- There are detailed allegations that the Ministry of Sport has orchestrated systematic doping and cover-ups.
Wada said officials in Russia were being stopped from testing athletes and threatened by security services.
Rune Andersen of the IAAF said Russian athletics had been "tainted by doping from the top level down."
"The systematic doping that has been ongoing in Russia - it's difficult to pick the clean athletes," he said.