Abdullah Abd al-Aal, 24, was shot in the stomach on the border east of Rafah in the southern Gaza Strip, a ministry spokesman said, in the first protests there since a deadly flare-up last weekend.
A further 13 demonstrators were shot during demonstrations in various places along the fractious border, the spokesman said.
The regular and often violent weekly protests, which began more than a year ago, were quieter than on normal Fridays, said an AFP correspondent on the scene.
An Israeli army spokeswoman said approximately 6,000 people took part in the demonstrations.
She cited "a number of explosions identified in the Gaza Strip, as well as a number of attempts to approach the fence" dividing the besieged coastal territory from Israeli territory.
Protesters are calling for Israel to end its more than decade-long blockade of Gaza.
This week's demonstrations were seen as a key test for a fragile truce agreed between Gaza's Islamist rulers Hamas and Israel.
Militants in Gaza fired hundreds of rockets at Israel last weekend, prompting Israeli forces to strike dozens of targets inside the Strip in response.
Over two days, 25 Palestinians were killed including at least nine militants, along with four Israeli civilians.
The truce agreement was announced by Palestinian factions early Monday.