The Spanish module for babel is responsible for this, probably following some directive of the authority on the language. There is an additional option for disabling it:
\usepackage[spanish,es-lcroman,es-nosectiondot]{babel}
See section 2.3 in the documentation for the Spanish option (texdoc spanish or here on TeXdoc.net).
There will be no need to act on \@seccntformat.
In order to keep the period after the chapter number (although I don't see why), patch the \@chapter command:
\usepackage{etoolbox}
\makeatletter
\patchcmd\@chapter{\numberline{\thechapter}}{\numberline{\thechapter.}}{}{}
\makeatother
To remove the word "Chapter" from the chapters lines (see this answer of egreg) and to add the dots to every line you can write the following lines in the preamble:
\makeatletter
\renewcommand\tocchapter[3]{%
\indentlabel{\@ifnotempty{#2}{\ignorespaces#2.\quad}}#3\dotfill%
}
\renewcommand{\tocsection}[3]{%
\indentlabel{\@ifnotempty{#2}{\ignorespaces#1 #2.\quad}}#3\dotfill%
}
\makeatother
Moreover, to insert the line with "Chapter" in the TOC, you can add the line
\addtocontents{toc}{\protect\contentsline{chapter}{Chapter}{}{}}
just after the \mainmatter and, to have a line with the word "Page" right-aligned at the beginning of the TOC, add this
\addtocontents{toc}{\protect\contentsline{chapter}{}{\textsc{\small Page}}{}}
just after the \frontmatter.
In other words, the following MWE:
\documentclass[10pt,twoside,titlepage,reqno]{amsbook}
\usepackage[a5paper]{geometry}
\geometry{papersize={5.375in,8.500in}}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[%hypertex,
unicode=true,
plainpages = false,
pdfpagelabels,
bookmarks=true,
bookmarksnumbered=true,
bookmarksopen=true,
breaklinks=true,
backref=false,
colorlinks=true,
linkcolor = blue,
urlcolor = blue,
citecolor = red,
anchorcolor = green,
hyperindex = true,
hyperfigures
]{hyperref}
\hypersetup{
pdftitle={Thermodynamics},
pdfauthor={Enrico Fermi},
pdfsubject={Unabridged, unaltered republication of 1937 edition.}
}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{lettrine}
\usepackage{type1cm}
\renewcommand{\LettrineTextFont}{\MakeUppercase}
\renewcommand{\thechapter}{\Roman{chapter}}
\usepackage{chngcntr}
\counterwithout{section}{chapter}
\author{\scshape{Enrico Fermi}}
\title{\Huge \MakeUppercase{Thermodynamics}}
\copyrightinfo{1936}{Enrico Fermi\\Copyright holder, Laura Fermi}
\makeindex
\makeatletter
\renewcommand\tocchapter[3]{%
\indentlabel{\@ifnotempty{#2}{\ignorespaces#2.\quad}}#3\dotfill%
}
\renewcommand{\tocsection}[3]{%
\indentlabel{\@ifnotempty{#2}{\ignorespaces#1 #2.\quad}}#3\dotfill%
}
\makeatother
\begin{document}
\frontmatter
\addtocontents{toc}{\protect\contentsline{chapter}{}{\textsc{\small Page}}{}}
\maketitle
\makeatletter
\@setcopyright
\makeatother
\chapter*{Preface}
\lettrine{T}{his} book originated in a course of lectures held at Columbia University,
New York, during the summer session of 1936.
Blablabla...
\begin{flushright}
\textsc{E. Fermi}
\end{flushright}
\tableofcontents
\chapter*{Introduction}
\lettrine{T}{hermodynamics} is mainly concerned with the transformations
of heat into mechanical work and the opposite transformations of mechanical work into heat.
Blablabla...
\mainmatter
\addtocontents{toc}{\protect\contentsline{chapter}{Chapter}{}{}}
\chapter{Thermodynamic Systems}\index{Thermodynamical systems}
\section{The state of a system and its transformations}\index{Thermodynamical state of a system}
The state of a system in mechanics is completely specified at a
given instant of time if the position and velocity of each mass
point of the system are given. For a system composed of a
number $N$ of mass-points, this requires the knowledge of
$6N$ variables.
Blablabla...
\subsection*{A system composed of a chemically defined homogeneous fluid}
\index{Chemically homogeneous systems}\index{Chemically homogeneous fluid}
We can make the following measurements on such a
system: the temperature $t$, the volume $V$, and the pressure $p$.
Blablabla...
\chapter{The First Law of Thermodynamics}
\section{The statement of the first law of thermodynarnics}
Some text...
\chapter{The Second Law of Thermodynamics}\label{chap:2ndLawOfThermo}
\printindex
\end{document}
gives the result:
Best Answer