Speaking on Friday, Bush accused the Syrian government of taking "disturbing steps" by arresting opposition activist Kamal al-Labwani "for serving as an advocate for democratic reform."
Bush criticised Syrian President Bashar al-Assad for delivering "a strident speech" that attacked both the Lebanese government and the integrity of Detlev Mehlis, the chief UN investigator of the 14 February killing of Rafiq al-Hariri, the former Lebanese prime minister.
"The government of Syria must do what the international community has demanded - cooperate fully with the Mehlis investigation and stop trying to intimidate and destabilise the Lebanese government.
"The government of Syria must stop exporting violence and start importing democracy," Bush said in a speech in Pennsylvania.
Bush's warning comes one day after al-Assad said his country would cooperate with the UN inquiry but would not sacrifice its own national interests.
Al-Assad: Syria will cooperate
with the UN al-Hariri inquiry
The US accuses Syria of allowing foreign fighters to cross its border into Iraq, supporting Palestinian and Lebanese fighters, and continued meddling in Lebanon.
Mehlis has until 15 December to complete his inquiry and report to the UN Security Council.
In an interim report last month, he criticised Syria for not cooperating properly with his mission.
That report also spoke of evidence pointing to Syrian and Lebanese involvement in al-Hariri's killing and said it would be hard to imagine how such a plot could have gone ahead without the knowledge of Syrian and Lebanese intelligence services.
Al-Assad dismissed the report as politically motivated and spoke bitterly about last month's Security Council resolution.