Kerry became the highest-ranking US official to meet President Abdel Fattah al-Sisi since he came to power earlier this month amid Egypt's rocky transition to democracy since the ousting of long-time leader Hosni Mubarak in 2011.
The top US diplomat's tour is also focused on trying to find a political solution in Iraq, where Islamic militants Sunday made new gains in an offensive that has triggered international alarm.
Kerry told a press conference he had emphasised in his "candid" talks with Sisi that the new government must uphold "the universal rights and freedoms of all Egyptians."
"We also discussed the essential role of a vibrant civil society, free press, rule of law and due process in a democracy."
Kerry's call for a free press in Egypt comes as a court on Monday is due to deliver its ruling in the trial of three Al-Jazeera journalists and 17 other co-defendants, accused of aiding the blacklisted Muslim Brotherhood of ousted president Mohamed Morsi.
Their trial, in which the three journalists are accused of "spreading false news" and links to the Brotherhood, has sparked an international outcry as rights activists fear Egypt could return to autocracy worse than under Mubarak.
"Obviously this is a critical moment of transition in Egypt," Kerry said earlier as he met new Foreign Minister Sameh Shoukri.
But he stressed "the United States remains deeply committed to seeing Egypt succeed."
"There is no question that Egyptian society is stronger when all of its citizens have a say and a stake in its success."
Since Egypt's first freely elected leader Morsi was toppled by Sisi in July 2013, a government crackdown on his supporters has left more than 1,400 people dead in street clashes and at least 15,000 jailed.
An US official told journalists travelling on the plane with Kerry that "the Egyptian government needs to have a very politically inclusive approach, which means that they need to include and find ways to reach out to the Muslim Brothers".
Kerry's visit comes a day after an Egyptian court confirmed death sentences for 183 Islamists, including Brotherhood chief Mohamed Badie.
US officials also revealed that $572 million (420 million euros) in aid, which had been frozen since October, was released to Egypt about 10 days ago after a green light from Congress. It will mainly go to pay existing defence contracts.
Washington said in April it planned to resume some of the annual $1.5 billion in mostly military aid to Cairo, including 10 Apache helicopter gunships for counterterrorism efforts in the Sinai Peninsula.
"The Apaches will come and they will come very, very soon," Kerry said minutes before he left Cairo and flew to Amman for further consultations on Iraq.
He insisted the US was not "responsible for what is happening in Iraq today" saying the ideology of the Islamic State of Iraq and Levant (ISIL) of "violence and repression is a threat not only to Iraq but to the entire region."
He urged all Iraq's neighbours, including Egypt, to call on Iraq's leaders "to rise above sectarian motivations and form a government that is united in its determination to meet the needs and speak to the demands of all of their people."
Egypt, the only Arab country with Jordan to have a peace treaty with Israel, has long been seen as a key strategic ally and a cornerstone to regional stability.
But the political turmoil has paralysed the nation, leaving it more concerned with domestic problems than regional matters despite the upheavals of the Arab Spring
Washington's concerns about Cairo include a new law controlling demonstrations, "the lack of space for dissent, mass trials and death sentences," said the official.
"We are concerned that some of the tactics they're using to address their security issues are polarising... they in some ways radicalise certain aspects of Egyptian society in ways that are not supportive of overall stability
