Your question is about ListPlot but the output is starting to resemble a BarChart. Can you perhaps use it in this particular instance?

pts = {1, 2, 3, 6, 4, 4};

BarChart[pts
 , ChartStyle -> {Directive[
    EdgeForm[{Black, Thin}], LightBlue]
   }
 , BarSpacing -> Medium
 , GridLines -> {Automatic, Automatic}
 , Epilog -> {AbsolutePointSize@Large, Darker@Cyan
   , Point@Transpose[{Range@Length@#, #} &@pts]
   }
 ]

Your question is about ListPlot but the output is starting to resemble a BarChart. Can you perhaps use it in this particular instance?

pts = {1, 2, 3, 6, 4, 4};

BarChart[pts
 , ChartStyle -> {Directive[
    EdgeForm[{Black, Thin}], LightBlue]
   }
 , BarSpacing -> Medium
 , GridLines -> {Automatic, Automatic}
 , PlotRange -> {{0.5, 6.6}, {-0.3, 6.7}}
 , Frame -> True
 , FrameTicks -> {{Range[0, Max@pts], None}
   , {Range@Length@pts, None}
   }
 , Epilog -> {AbsolutePointSize@Large, Darker@Cyan
   , Point@Transpose[{Range@Length@#, #} &@pts]
   }
 ]