Clearing floats in LaTeX

With some care you can clear wrapfig's settings so the later text is not indented, although you need to add the extra space to compensate. (If you were doing this often you could automate the measurement of the image and the size of the paragraph, but not really worth it for a hand written docuemnt, only for an automated workflow)

Note wrapfig, even with L rather than l isn't really a latex float so the question title is a bit misleading. I used \section as the custom definition seemed to miss all the important code for a section heading (controlling page breaks after the heading for example)

Also added some missing % and used \centering (which adds no vertical space) rather than center with negative space adjustments.

\documentclass[twoside]{memoir}
\usepackage[papersize={8.5in,11in}, vmargin=0.5in, outer=1in, inner=0.5in, includehead, includefoot]{geometry}
\usepackage{color}
\usepackage{fontspec}
\usepackage{newunicodechar}
\usepackage[nopar]{lipsum}% http://ctan.org/pkg/lipsum
\usepackage{etoolbox}% http://ctan.org/pkg/etoolbox
\usepackage[none]{hyphenat}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{placeins}
\graphicspath{ {images/} }

\makeatletter

\newcommand\floatingimage[4]{%
  \begin{wrapfigure}{#3}{0.25\textwidth}%
\centering
      \includegraphics[width=0.25\textwidth]{#1}%

    \legend{#4}%
  \end{wrapfigure}%
}

% Remove 'Figure 1' text


\makeatother

\setlength{\parindent}{0pt}%
\nonzeroparskip

\begin{document}
\chapterstyle{scp}
\chapter{Example}
\floatingimage{example-image-a}{800}{R}{example-image-a}
\section{Section 1}
\lipsum[1-1]

\section{Section 2}
\lipsum[2-2]

\floatingimage{example-image-b}{150}{l}{example-image-b}
\lipsum[3-3]

\makeatletter
\WF@@everypar{}
\vspace{\baselineskip}
\makeatother

\section{Section 3}
\lipsum[4-4]
\end{document}