The acceleration due to gravity is vertically downward if we take horizontal component then angle between horizontal and vertical is 90.hence horizontal component = g*cos(angle between horizontal and vertical)
                 = g*cos90
                 = 0
hence horizontal acceleration is zero.

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As per Newtons' law a body continues to be at rest or move at uniform speed in the absence of an external force.

On earth's surface only significant force on a projectile in air, is the Earth's magnetic attraction force. This is directed perpendicular to the surface and vertically downwards. There is no force in the horizontal direction, so there is no acceleration horizontally.