how to round corners of shaded boxes

Perhaps tcolorbox makes things easier.

https://ctan.org/pkg/tcolorbox?lang=en

% !TeX TS-program = pdflatex

documentclass{memoir}
usepackage{titling}

usepackage{algorithm,algpseudocode}
usepackage{bm}  % bold greek variables (boldmath)
usepackage{tikz}
usetikzlibrary{positioning}  % use: above left of, etc
usetikzlibrary{calc}

% Required for the ToDo list.
usepackage{ifthen}

usepackage[protrusion=true,expansion=true, kerning]{microtype}
usepackage{enumitem}

usepackage{booktabs}
usepackage{lipsum}
usepackage{wrapfig}

usepackage{tcolorbox}

newcommand{secref}[1]{Section~ref{#1}}
newcommand{chapref}[1]{Chapter~ref{#1}}
newcommand{eqref}[1]{Equation~(ref{#1})}
newcommand{figref}[1]{Figure~ref{#1}}
newcommand{tabref}[1]{Table~ref{#1}} 

begin{document}

definecolor{shadecolor}{RGB}{176,224,230}%{246,243,226}
%setlengthOuterFrameSep{0pt}
%begin{shaded*}
%    % medskip 
%    textcolor{gray}{}
%    textcolor{gray}{textbf{Chapter 1}}
%    textcolor{dark_red}{textbf{SUMMARY: }}
%    medskip    
%    Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text, 
%end{shaded*}
%
%
%
%setlengthOuterFrameSep{0pt}
%begin{shaded*}
%    % medskip 
%    textcolor{gray}{}
%    textcolor{gray}{textbf{Chapter 2}}
%    textcolor{dark_red}{textbf{SUMMARY: }}
%    medskip    
%    Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text,Some text, 
%end{shaded*}


begin{tcolorbox}
    lipsum[3]
end{tcolorbox}


begin{tcolorbox}[colback=white, colframe=blue!20, arc=10pt, title=color{gray} textbf{Chapter 1} color{red}  textbf{SUMMARY:}]
    lipsum[1]
end{tcolorbox}

end{document}

You did not include the environment shaded (perhaps from framed ?) in your code, so I commented these lines.