1

I tried to remove it, to empty it but nothing works until now ...

\documentclass[11pt,a4paper,roman]{moderncv} 
\usepackage[spanish,es-lcroman]{babel}
\usepackage{ragged2e}
\setlength\parindent{24pt}

\moderncvstyle{classic}                           
\moderncvcolor{green}

\usepackage[utf8]{inputenc}                      
\name{Antoine}{Coppin}
\title{Lettre de motivation }                          
\address{Somewhere}{75000 Paris}
\phone[mobile]{06-00-00-00-00}                  
\recipient{Monsieur le général sous-directeur\\du recrutement de l'Armée de terre}{}
\date{24 décembre 2017}
\opening{}
\closing{.}
%\enclosure[Pièce-jointe]{CV}          % use an optional argument to use a string other than "Enclosure", or redefine \enclname
\makelettertitle

% permet de justifier
\justifying
\hspace{4ex}Agé de 24 ans, actuellement en Master 2 Recherche en Intelligence Artificielle à l’Université Paris-Dauphine ...

\makeletterclosing

\end{document}
2

Under your specific case, the letter closing is defined like this:

\renewcommand*{\makeletterclosing}{
  \@closing\\[3em]%  <--------------------- insert closing
  {\bfseries\@firstname~\@lastname}%
  \ifthenelse{\isundefined{\@enclosure}}{}{%
    \\%
    \vfil%
    {\color{color2}\itshape\enclname: \@enclosure}}%
    \vfil}

The closing (stored inside \@closing) is inserted, followed by a line-break of length 3em. We can remove this by adding the following anywhere in your preamble:

\makeatletter
\AtBeginDocument{\def\@closing\\[#1]{}}
\makeatother

The above redefinition of \@closing removes the entire first line highlighted above (gobbling the sequence \@closing\\[<anything>]).

enter image description here


You can, of course, also replicate the output using article instead:

enter image description here

\documentclass[11pt]{article}

\usepackage[margin=1.5in]{geometry}
\usepackage[spanish,es-lcroman]{babel}
\usepackage[utf8]{inputenc}
\usepackage{xcolor,fontawesome}

\pagestyle{empty}
\setlength\parindent{24pt}

\begin{document}

\hfill
{\color{black!80}\slshape\begin{tabular}{ r @{} }
  \bfseries\upshape Antoine Coppin \\
  Somewhere \\
  75000 Paris \\
  \textup{\faPhone} 06-00-00-00-00
\end{tabular}}

\bigskip

\noindent
{\bfseries\begin{tabular}[t]{ @{} l }
  Monsieur le général sous-directeur \\
  du recrutement de l'Armée de terre
\end{tabular}} \hfill
24 décembre 2017

\bigskip

Agé de 24 ans, actuellement en Master 2 Recherche en Intelligence Artificielle à l’Université Paris-Dauphine ...

\bigskip

\noindent
\textbf{Antoine Coppin}

\end{document}

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