Diagonal Arrows Under an Equation

Since you want to avoid tikzmark, you can use the standard picture environment. It may require some fine-tuning for placement.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage{multicol,array,tabularx,xcolor,enumitem}
\usepackage{amsmath,amsfonts,amssymb,mathrsfs}
\pagestyle{empty}

\definecolor{modra}{cmyk}{1,0,0,0} %%%%DEFINCIE%%%%%BARVY
\definecolor{cerna}{cmyk}{0,0,0,1}
\definecolor{bila}{cmyk}{0,0,0,0}
%
%\color{modra}
%

\begin{document}

\setlength{\unitlength}{1mm}
\begin{picture}(100,80)
\put(0,80){
\begin{minipage}[t]{\columnwidth}
\begin{align*}
   S & = \rho \cdot \dfrac{a + b +c}{2}     & S & = \dfrac{a \cdot b}{2} \\
   \rho \cdot \dfrac{a + b +c}{2} & = \dfrac{a \cdot b}{2}  \qquad | \cdot 2 & & \\ 
   18 (a +b+ c ) & = a \cdot b          &  a  & = b  \\
   18 (a + a + c ) & = a \cdot a            &  c^2 & = a^2 + b^2 \\
   18 (2a + c ) & = a^2             &  c^2 & = a^2 + a^2 = 2a^2 \\
   36 a + 18 c  & = a^2         &  c & = \sqrt{2a^2} = a \sqrt{2} \\
   36 a + 18  a \sqrt{2}  & = a^2           &  & \\
   a^2  - 36 a - 18  a \sqrt{2} & = 0       &  & \\
   a \left( a   - 36  - 18   \sqrt{2} \right) & = 0     &  & \\
\end{align*}
\end{minipage}
\thicklines
\put(-114,-67){\color{red}\vector(-1,-1){7}}
\put(-98,-67) {\color{red}\vector(1,-1){7}}
}
\end{picture}

\end{document}

Here is a simple solution with tikzmark:

• using \tikzmarknode
• commenting out unused parts and colors

\documentclass[11pt,a4paper]{article}
%\usepackage[utf8]{inputenc}
%\usepackage[czech]{babel}
%\usepackage[T1]{fontenc}
%\usepackage{pgfplots}
%\usepackage{multicol,array,tabularx,xcolor,enumitem}
\usepackage{amsmath,amsfonts,amssymb,mathrsfs}
\pagestyle{empty}
% ~~~ let's use Tikz ~~~~~~~~~~~~~~~
\usepackage{tikz}
\usetikzlibrary{tikzmark}   % to draw over the equations
\usetikzlibrary{arrows.meta}% for nicer arrow tips
%
%
%
%
%\definecolor{modra}{cmyk}{1,0,0,0} %%%%DEFINCIE%%%%%BARVY
%\definecolor{cerna}{cmyk}{0,0,0,1}
%\definecolor{bila}{cmyk}{0,0,0,0}
%
%\color{modra}
%
\begin{document}
\begin{align*}
   S & = \rho \cdot \dfrac{a + b +c}{2}     & S & = \dfrac{a \cdot b}{2} \\
   \rho \cdot \dfrac{a + b +c}{2} & = \dfrac{a \cdot b}{2}  \qquad | \cdot 2 & & \\ 
   18 (a +b+ c ) & = a \cdot b          &  a  & = b  \\
   18 (a + a + c ) & = a \cdot a            &  c^2 & = a^2 + b^2 \\
   18 (2a + c ) & = a^2             &  c^2 & = a^2 + a^2 = 2a^2 \\
   36 a + 18 c  & = a^2         &  c & = \sqrt{2a^2} = a \sqrt{2} \\
   36 a + 18  a \sqrt{2}  & = a^2           &  & \\
   a^2  - 36 a - 18  a \sqrt{2} & = 0       &  & \\
   \tikzmarknode{A}{a} \left( a   - \tikzmarknode{nmb}{36}  - 18   \sqrt{2} \right) & = 0     &  & \\
% \Large \swarrow \qquad \searrow
\end{align*}

\begin{tikzpicture}[
    remember picture, overlay,          % needed with tikzmark
    arr/.style={red,->,line width=2pt}, % arrow
    >={Stealth},                        % other tip
]
    \draw[arr] ([yshift=-2mm]A.south west) -- +(-135:1);
    \draw[arr] ([yshift=-2mm]nmb.south) -- +(-45:1);
\end{tikzpicture}
%
And so we continue \dots
%
\end{document}

I would instead recommend tikzmark, but here is a try, using zero sized boxes and a tabular to position the arrow and an optional content

\NewDocumentCommand{\annotate} { s m O{} } {%
  \IfBooleanTF {#1}
    {% point right
      #2%
      \raisebox{-\height}[0pt][0pt]
        {%
          \rlap
            {%
              \begin{tabular}{@{} r @{}}
                \Large $\searrow$ \\
                \clap{#3}
              \end{tabular}%
            }%
        }
    }
    {% point left
      \raisebox{-\height}[0pt][0pt]
        {%
          \llap
            {%
              \begin{tabular}{@{} l @{}}
                \Large $\swarrow$ \\
                \clap{#3}
              \end{tabular}%
            }%
        }
      #2%
    }%
}

With a star, the arrow points on the left, without, on the right. The optional argument is centered under the arrow.

Example:

\documentclass[11pt,a4paper]{article}

\usepackage[czech]{babel}
\usepackage[T1]{fontenc}
\usepackage{amsmath,amsfonts,amssymb,mathrsfs}
\usepackage{graphicx}

\NewDocumentCommand{\annotate} { s m O{} } {%
  \IfBooleanTF {#1}
    {% point right
      #2%
      \raisebox{-\height}[0pt][0pt]
        {%
          \rlap
            {%
              \begin{tabular}{@{} r @{}}
                \Large $\searrow$ \\
                \clap{#3}
              \end{tabular}%
            }%
        }
    }
    {% point left
      \raisebox{-\height}[0pt][0pt]
        {%
          \llap
            {%
              \begin{tabular}{@{} l @{}}
                \Large $\swarrow$ \\
                \clap{#3}
              \end{tabular}%
            }%
        }
      #2%
    }%
}

\begin{document}


Dummy text before\dots
\begin{align*}
   S & = \rho \cdot \dfrac{a + b +c}{2}     & S & = \dfrac{a \cdot b}{2} \\
   18 (a +b+ c ) & = a \cdot b          &  a  & = b  \\
  \dots\\
   \annotate{a} \left( a   - \annotate*{36}  - 18   \sqrt{2} \right) & = 0     &  & \\
\end{align*}
\dots and after.

\bigskip

Dummy text before\dots
\begin{align*}
   S & = \rho \cdot \dfrac{a + b +c}{2}     & S & = \dfrac{a \cdot b}{2} \\
   18 (a +b+ c ) & = a \cdot b          &  a  & = b  \\
  \dots\\
   \annotate{a}[this is $a$] \left( a   - \annotate*{36}[this is 36]  - 18   \sqrt{2} \right) & = 0     &  & \\
\end{align*}
\dots and after.

\end{document}