The USS George H W Bush has been a leading part of the US bombing campaign in Iraq and Syria, primarily against the Islamic State group but on occasion targeting Syrian government forces and their allies fighting US-backed groups.
The 330-metre ship carries between 70 and 90 aircraft at a time, including a fleet of F-18 fighter jets which are used to strike IS targets in Mosul, Raqa and elsewhere.
The Syrian regime is also fighting IS, as well as battling US allies in other parts of the war-ravaged country.
McCall said an aircraft from the ship was the one that had downed a Syrian warplane on June 18, after the US military had issued a series of warnings.
He admitted there had been "a lot of tension" with "pro- regime forces" in parts of Syria.
"We have actually flown in support of some of those actions -- to include air-to-surface engagements -- so that is absolutely something we are paying a lot of attention to."
The US carried out its first direct military action against Syria's regime in April, firing 59 Tomahawk cruise missiles at the Shayrat military airport in response to an alleged chemical weapons attack.
Last week, the White House said President Bashar al-Assad was potentially preparing a new chemical weapons attack.
Pennington said between 20 and 25 missions took off a day at peak times.
He said the forces had played a crucial role in pushing back IS in Iraq, with the militants losing control of the vast majority of their former stronghold city of Mosul.
"The battle is coming to a conclusion in Mosul, although I think the last few things will be tough given the narrow streets and densely populated (neighbourhoods)."
The US military reported last month that 484 civilians have been killed during attacks on IS targets since mid-2014, including 105 who died when a US aircraft dropped a bomb on an IS sniper in Mosul on March 17.
McCall said everything possible was being done to prevent such deaths.
"Minimising civilian casualties on the ground is a very important part of what we do.
"We have had aviators that have spotted what they believed to be civilian personnel on the ground or just they are not sure who those are.
"In all those cases they have been able to communicate back on the radios to our folks on the ground and either stop the attack from happening or wait until that confusion could be cleared up."
The ship's arrival marked the first time since 2000 that an American aircraft carrier has visited Israel.
Israeli Prime Minister Benjamin Netanyahu boarded on Monday along with the US ambassador to Israel David Friedman.
Netanyahu hailed the ship's role in taking on IS.
"This carrier has served as a base for flights against the Islamic State and a war against ISIS should be seen for what it is -- a clash between good and evil," Netanyahu told crew members and journalists aboard.
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