At a summit with Russian President Vladimir Putin in Finland, Mr Trump contradicted US intelligence agencies, saying Russia had no reason to meddle.
The top Republican in Congress, House Speaker Paul Ryan, said Mr Trump must see that "Russia is not our ally".
Republican Senator John McCain said it was a "disgraceful performance".
Mr Putin denied the interference claim.
The US and Russian presidents held nearly two hours of closed-door talks in the Finnish capital Helsinki on Monday.