The action by the US, France and the UK targeted the regime’s chemical weapons in response to an alleged gas attack in the city of Douma a week ago.
But both rebels and opposition politicians say the Western allies should have also hit Mr Assad’s conventional weapons – which they say will now be used to target more people.
“Maybe the regime will not use chemical weapons again, but it will not hesitate to use other weapons… such as barrel bombs,” said leader of the opposition Nasr al-Hariri on Twitter.
Rebels said they were bracing themselves for revenge action in territory in the north-west of the country which a senior Iranian official had earlier indicated could be the next target, Reuters reported.
“I am expecting an escalation by the regime against civilians in Idlib and in the areas of northern Syria and the liberated areas, because the regime always takes revenge on civilians,” one senior rebel in Hama province told the news agency.
He added: “More was expected from the American strike to affect the path of the war and to curb Assad’s crimes.”
Another rebel, Mohammad Alloush of the Jaish al-Islam group, was quoted by the AFP news agency as saying: “Punishing the instrument of the crime while keeping the criminal – a farce.”
The fears appeared to be confirmed within hours of the strikes, as Mr Assad declared the action – carried out in the early hours of Saturday morning – would only strengthen his resolve to continue “fighting and crushing terrorists in every inch of the nation”.
Earlier, the Syrian government had posted a video online appearing to show the president, unperturbed, arriving for work in the capital Damascus.
Footage shows Mr Assad wearing a suit and tie and carrying a briefcase as he walks into a marble-floored building.
It was captioned “the morning of resilience” and circulated on the presidency’s social media feeds.
The defiance was supported by key regional ally Iran.
World news in pictures
President Hassan Rouhani said the country would remain standing by Syria and expressed his “confidence that this aggression would not weaken the determination of the Syrian people in its war against terrorism”.
A few hours earlier loud explosions jolted Damascus and the sky turned orange as Syrian air defence units fired surface-to-air missiles in response to three waves of military strikes designed to punish Mr Assad for his alleged use of chemical weapons last week.
But Syria claims 71 out of 103 missiles were shot out the sky. Shortly after the attack ended, vehicles with loudspeakers drove around the streets of Damascus blaring nationalist songs.