Yemeni activist and 2011 winner of the Nobel Peace Prize, Tawakkol Karman, said it would be an "awful crime" if the claims of his death were true.
"Killing him is like killing us. This policy is just a terror policy. There's no difference between the state terror and other terror actions," she added.
Khashoggi went to the consulate to obtain official documents ahead of his marriage to his Turkish fiancee. Turkish police quickly said he never left the building as there was no security footage on his departure.
The consulate rejected the claims that the journalist was killed there as "baseless".
"We will allow them to enter and search and do whatever they want to do. If they ask for that, of course, we will allow them. We have nothing to hide," Prince Salman said in an interview published on Friday.
Khashoggi had been critical of some of the crown prince's policies and Riyadh's intervention in the war in Yemen in Arab and Western media.
"As of now, I would say Mohammed bin Salman is acting like Putin. He is imposing very selective justice. The crackdown on even the most constructive criticism - the demand for complete loyalty with a significant 'or else' - remains a serious challenge to the crown prince's desire to be seen as a modern, enlightened leader," he wrote.
"We hope to have results very quickly," Erdogan said. "I am waiting, with high hopes."
Erdogan also said police officers were examining CCTV footage of entrances and exits at the consulate and Istanbul airport.
But Haytham Abokhalil, an Istanbul-based Egytian presenter at Al-Sharq TV, said supporters wanted "strong action" from Erdogan.
Abokhalil and conflict expert Mohamed Okda, both friends of Khashoggi's, demanded explanations from Prince Mohammed over the disappearance.
"We demand the international community to pressure Saudi Arabia and Mohammed bin Salman to tell us exactly what happened inside the consulate... and to tell us every detail of who was involved in this crime," Okda told AFP.
Saudi Arabia launched a modernisation campaign following Prince Mohammed's appointment as heir to the throne with moves such as lifting a ban on women driving.
But the ultra-conservative kingdom, which ranks 169th out of 180 on RSF's World Press Freedom Index, has been strongly criticised over its intolerance of dissent with dozens of people arrested including intellectuals and Islamic preachers.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)