Gravity is a strange force. It's powerful enough to keep a giant chunk of rock rotating around the earth, but it's not powerful enough to prevent me from jumping into the air.
It's clear gravity isn't caused by the Sun. Otherwise, we'd get sucked to the sun instead of staying on Earth. Likewise, Jupiter wouldn't have more gravity than us because their farther away from the sun and travel slower around it. However, their gravity is strong enough to attract dozens of moons and would be crushing if we were to enter Jupiter's atmosphere.