No he didn't. WWI and its related military spending ended 14 months before the depression of 1920 began. Military spending was still included in the 1919 Federal budget which was $18.9b. The 1920 budget shows that military spending had already been cut out when the federal budget was a mere $6.8b (source: http://www.usgovernmentspending.com/year1918_0.html#usgs302).
Harding further cut spending in March 1921 (2 1/2 years after the war had ended) ultimately reducing the federal budget by 50% in two years. These were cuts in general government spending, not war spending.
This is completely off topic and I don't really want to go down this road. Even using the figures on that site, Harding chopped the federal budget from USD 6.8 billion to USD 3.8 billion by 1922. Out of this, USD 1.3 or nearly 44% of the cut came from a drop in military spending. The other big change is under the balance heading which I have no idea what it relates to. But you can see yourself that the spending on the other headline areas did not change to any great extent.
You also credit him with chopping income tax rates but what you are leaving out is that tax rates were at extremly high levels to pay for WWI. As far as I can remember people were paying nearly 60% rates on income over USD 100,000 by 1920. By 1925, Harding and then Coolridge had brought these down to more respectable levels (about 20-25% I think). However these rates were still a lot higher the rates that were seen in the US before WWI.
Like I say, completely off topic!