"Stair-ify" a string

Pyth, 63 bytes

V_Q aY?}rN0"aeiou"=hZ?}N"     "=tZZ;Jh.mbYKh.MZYjC.b++*d+JNY*dK_YQ
                         ^^^^^
                         |||||
                         |tabs
                        space

The spaces in the middle is actually a single tab character, but StackExchange renders it as four spaces.

Try it online!

Python 2, 141 137 bytes

def S(s,l=[0]):
 for c in s:l+=[l[-1]-(c in"aeiouAEIOU")+(c<"!")]
 for h in sorted(set(l)):print"".join([" ",c][i==h]for i,c in zip(l,s))

JavaScript (Firefox 30-57), 151 bytes

s=>[...s].map((c,i)=>r[c<'!'?n++:/[AEIOU]/i.test(c)?n--:n][i]=c,n=s.length,r=[for(_ of s+s)[]])&&[for(a of r)if(s=[for(c of a)c||' '].join``)s].join`\n`

Where \n represents the literal newline character.