The Singapore company announced on Friday that it would no longer bid after rival Dubai Ports World made a higher offer.
Dubai Ports World last month raised its offer to $6.9 billion for the company, trumping a $6.3 billion offer from PSA, a unit of Singapore's government-owned investment company, Temasek Holdings.
A company statement said: "PSA has decided not to increase its offer and will therefore no longer pursue the acquisition of P&O. For PSA, to pay more than this price would not be compatible with commercial business sense and PSA's future success."
Founded in 1837, P&O carried cargo throughout the British empire in its heyday to Sydney, Calcutta, Singapore, Hong Kong and other colonies.
The addition of P&O's 29 ports around the world would have helped PSA gain ground on the world's biggest operator, Hutchison Whampoa of Hong Kong.
In a statement, P&O Chairman Sir John Parker thanked PSA's leadership "for their professionalism and courtesy throughout this process."
"The P&O Directors recommend unanimously the DP World revised offer and look forward to putting it to Stockholders," the statement said. "The combination of P&O and DP World has compelling strategic logic and will create significant opportunities for both businesses and their employees."