Mr Abiy said he was not aware of the criteria used to select him.
He was credited for his move to make peace with neighbouring Eritrea.
Mr Trump said last week that he had “saved a country” from a big war, a possible reference to his work on another dispute involving Ethiopia.
Ethiopia and Eritrea fought a bitter border war from 1998-2000, which killed tens of thousands of people.
Although a ceasefire was signed in 2000, the neighbours technically remained at war until July 2018, when Mr Abiy and Eritrea’s President Isaias Afwerki signed a peace deal.
For two decades, the long border was closed, dividing families and making trade impossible.
He told reporters during a press briefing in South Africa that he was “not working for the prize” but for “peace in the region”.
“I don’t have any clue about the criteria, how the Nobel committee selects an individual for the prize,” he said.
“If President Trump complained it must go to Oslo not to Ethiopia,” he added.
He told supporters at a political rally last week that he had helped avert a war in “a country” and expressed surprise that his efforts were not recognised.
“I made a deal, I saved a country, and I just heard that the head of that country is now getting the Nobel Peace Prize for saving the country. I said: ‘What, did I have something to do with it?’ Yeah, but you know, that’s the way it is,” he said.
A Twitter account run by the Democrats in Congress teased Mr Trump saying he seemed to not know why Mr Abiy won the peace prize.
The tweet said the president had confused his meeting last year in November with representatives from Egypt, Ethiopia and Sudan, over a dispute about the use of the Nile waters, as worth of a Nobel prize.