On Friday The New York Times commented that "Europe seemed to be living down to expectations" and the German Speigel Online said "Europe counts on nobodies".
"The expectations are so low that Mr Van Rompuy and Ms Ashton can only be positive surprises. No wonder there is no enthusiasm about Europe," the publication said.
Jacques Reland, head of European research at the Global Policy Institute, told Al Jazeera that the appointments were underwhelming.
"After all the ballyhoo about the treaty helping to make Europe a stronger voice in the world etc, the first appointments were really a bit disappointing
"The job description was definitely not someone to 'stop the traffic' like Tony Blair [former British prime minister], but more a person able to build a consensus between the different views of the leaders."
Critics have also pointed to Ashton's lack of foreign policy experience and that she has never been elected to public office.
But she responded on Friday saying she intended to prove that she was the best person for the job.
"I think for quite a few people they would say that I am the best person for the job and I was chosen because I am, but I absolutely recognise there are a number of candidates around, all of whom would have been extremely good, extremely able," she told BBC radio.
She added that she was not unknown on the global stage as, in her role as trade commissioner, she had worked with ministers across the world at vital trade talks.
The appointment of Van Rompuy, who has previously said that he viewed the EU as a Christian club with no room for Turkey and its mainly Muslim population has raised concern from the country.
Onur Oymen, a Turkish MP, his country was "not very optimistic about the future of our relations during his presidency".
But many world leaders have welcomed the two new EU heads, with Gordon Brown, the British prime minister, welcoming the decision to appoint Ashton to the role.
Hillary Clinton, the US secretary of state, hailed the appointments as "a milestone for Europe and for its role in the world," and described Ashton as "my new counterpart."
"I look forward to working closely with them to strengthen and broaden our partnership - from achieving stability in Afghanistan to securing Iranian compliance with its nonproliferation obligations and promoting a comprehensive peace in the Middle East."
"With the appointment of these distinguished leaders, I am more confident than ever that together we can build a more peaceful and prosperous world."
The Lisbon Treaty which created the new post was vague on its powers and obligations, saying simply that the president should "drive forward" the EU's work.
Van Rompuy will serve a two-and-a-half year term, which is renewable once.