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]
       }
     ]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/zj54Q.png