The Palestinians took part in an action announced by Hamas in protest against Israel's slowing down of financial support from Qatar, writes AP.
Hamas also stressed that the day of protest coincided with the 52nd anniversary of a fire in the al-Aqsa Mosque.
According to the news agency AFP, they threw firebombs and burning car tires over the border fence against Israel.
The Israeli border forces responded with tear gas and sharp shots, writes The Times of Israel.
One of the seriously injured Palestinians is a 13-year-old boy who was shot in the head, according to the Gaza Ministry of Health.
The Israeli soldier is said to have been shot with a pistol.