I tried Reduce, which is more powerful than Resolve and got a message that it can't be done with Reduce.
In[3]:= Reduce[
Exists[{d}, (v + b (1 - r) (d)^0.5 -
d - ((v + b (1 - r) (d)^0.5 - d)^2 - v^2)^0.5) (v +
b (1 + r) (d)^0.5 -
d + ((v + b (1 - r) (d)^0.5 - d)^2 - v^2)^0.5) -
4 (v/2 + r^2 b^2/8)^2 == 0 && d > 0 &&
d < v && ((3/4 b^2 r^2 < v < 3/4 b^2 &&
r < (8 v*b^2 + b^4)/(4 v + b^4)) || (v > 3/4 b^2 &&
r < ((1 + b^2) v + b^4/4)/(4 v + b^4))) && 0 < b < 1 && v > 0 &&
r > 0], {d, v, b, r}, Reals]
During evaluation of In[3]:= Reduce::nsmet: This system cannot be solved with the methods available to Reduce.
Out[3]= Reduce[\!\(
\*SubscriptBox[\(\[Exists]\), \({d}\)]\((\(-4\)\
\*SuperscriptBox[\((
\*FractionBox[\(
\*SuperscriptBox[\(b\), \(2\)]\
\*SuperscriptBox[\(r\), \(2\)]\), \(8\)] +
\*FractionBox[\(v\), \(2\)])\), \(2\)] + \((\(-d\) + b\
\*SuperscriptBox[\(d\), \(0.5`\)]\ \((1 - r)\) + v -
\*SuperscriptBox[\((\(-
\*SuperscriptBox[\(v\), \(2\)]\) +
\*SuperscriptBox[\((\(-d\) + b\
\*SuperscriptBox[\(d\), \(0.5`\)]\ \((1 - r)\) +
v)\), \(2\)])\), \(0.5`\)])\)\ \((\(-d\) + b\
\*SuperscriptBox[\(d\), \(0.5`\)]\ \((1 + r)\) + v +
\*SuperscriptBox[\((\(-
\*SuperscriptBox[\(v\), \(2\)]\) +
\*SuperscriptBox[\((\(-d\) + b\
\*SuperscriptBox[\(d\), \(0.5`\)]\ \((1 - r)\) +
v)\), \(2\)])\), \(0.5`\)])\) == 0 && d > 0 &&
d < v && \((\((
\*FractionBox[\(3\
\*SuperscriptBox[\(b\), \(2\)]\
\*SuperscriptBox[\(r\), \(2\)]\), \(4\)] < v <
\*FractionBox[\(3\
\*SuperscriptBox[\(b\), \(2\)]\), \(4\)] && r <
\*FractionBox[\(
\*SuperscriptBox[\(b\), \(4\)] + 8\
\*SuperscriptBox[\(b\), \(2\)]\ v\), \(
\*SuperscriptBox[\(b\), \(4\)] + 4\ v\)])\) || \((v >
\*FractionBox[\(3\
\*SuperscriptBox[\(b\), \(2\)]\), \(4\)] && r <
\*FractionBox[\(
\*FractionBox[
SuperscriptBox[\(b\), \(4\)], \(4\)] + \((1 +
\*SuperscriptBox[\(b\), \(2\)])\)\ v\), \(
\*SuperscriptBox[\(b\), \(4\)] + 4\ v\)])\))\) && 0 < b < 1 && v > 0 &&
r > 0)\)\), {d, v, b, r}, Reals]