"If Russia wants to be absolved of responsibility for future attacks, Vladimir Putin needs to enforce commitments, to dismantle Assad's chemical weapons arsenal for good, and to get fully engaged with the UN peacekeeping progress," he added.
Fallon reiterated Britain's position that Assad should quit.
"Someone who uses barrel bombs and chemicals to kill his own people simply cannot be the future leader of Syria," he wrote.
Assad's future role is a key sticking point -- the rebels and their international backers demand that he must step down.
But Assad refuses to budge. His key ally in Moscow has backed him to the hilt against the rebels.
"Today we call on all parties to get back to the table and get a deal done," the British defence minister said.
"That deal must lead to a representative government in which Assad will play no part".
He also once again offered Britain's support to the United States for its decision on Friday to fire 59 Tomahawk cruise missiles at the airfield located near Homs in central Syria.
The move was in response to the suspected chemical weapons attack on the rebel-held town of Khan Sheikhun which killed 87 people according to the Syrian Observatory of Human Rights.
"Given repeated Russian blocking in the UN security council, the US was determined to act," Fallon said, adding President Trump made "the right call by resorting to careful and narrowly focused military action".
Russia has criticised the US military intervention as a "gross... violation of international law".
It also slammed the British foreign secretary Boris Johnson's decision to cancel a scheduled visit to Russia, claiming Britain has "no real influence" internationally.
Johnson decided to abandon his visit to Moscow on Monday, saying he deplored "Russia's continued defence of the Assad regime".
He argued it would be best for US Secretary of State Rex Tillerson to deliver a "clear and coordinated message" to the Russians during his own visit to Moscow later in the week.
Priti Patel, Britain's Secretary of State for international development also announced on Sunday that the country was to increase its support of medical aid in Syria in the wake of the attack.
The new funding totalling £7 million ($8.6 million, 8.2 million euros) would be donated to the World Health Organization (WHO) and two undisclosed non-governmental organisations.
According to the statement, the allocation was to be used to provide clean water for 500,000 people, medicines and medical supplies to over 400 clinics and hospitals, and training for more than 400 Syrian health workers.
Britain donated £10.7 million to WHO in the 2016/17 financial year.