Russia had agreed to ensure Syria's chemical weapons stockpiles were destroyed - and its failure to do this enabled the attack, he said.
G7 foreign ministers are preparing to meet in Italy later on Monday.
Talks will focus on how to increase pressure on Russia to distance itself from Syria's President Bashar Al-Assad.
On Tuesday, Mr Tillerson will continue from the G7 to Moscow, where he will meet his Russian counterpart, Sergei Lavrov.
Russia is the Syrian government's main ally, and helped facilitate a 2013 agreement to destroy Syria's chemical arsenal.
The suspected chemical attack on Khan Sheikhoun last Wednesday left 89 people dead.
In response, the US fired 59 missiles at a Syrian airbase on Friday.
Syria has denied using any chemical agents, and Russia says the US gave no evidence Syria has chemical weapons.