Apart from the tkinter modules, we will also need the calendar and datetime modules from the Standard Library. They will help us to model and interact with the data held by the widget.

The widget header displays a couple of arrows to move the current month back and forth, based on Ttk styling options. The main body of the widget consists of a ttk.Treeview table, with a Canvas instance that highlights the selected date cell:

import calendar
import datetime
import tkinter as tk
import tkinter.ttk as ttk
import tkinter.font as tkfont
from itertools import zip_longest

class TtkCalendar(ttk.Frame):
    def __init__(self, master=None, **kw):
        now = datetime.datetime.now()
        fwday = kw.pop('firstweekday', calendar.MONDAY)
        year = kw.pop('year', now.year)
        month = kw.pop('month', now.month)
        sel_bg = kw.pop('selectbackground', '#ecffc4')
        sel_fg = kw.pop('selectforeground', '#05640e')

        super().__init__(master, **kw)

        self.selected = None
        self.date = datetime.date(year, month, 1)
        self.cal = calendar.TextCalendar(fwday)
        self.font = tkfont.Font(self)
        self.header = self.create_header()
        self.table = self.create_table()
        self.canvas = self.create_canvas(sel_bg, sel_fg)
        self.build_calendar()

    def create_header(self):
        left_arrow = {'children': [('Button.leftarrow', None)]}
        right_arrow = {'children': [('Button.rightarrow', None)]}
        style = ttk.Style(self)
        style.layout('L.TButton', [('Button.focus', left_arrow)])
        style.layout('R.TButton', [('Button.focus', right_arrow)])

        hframe = ttk.Frame(self)
        btn_left = ttk.Button(hframe, style='L.TButton',
                              command=lambda: self.move_month(-1))
        btn_right = ttk.Button(hframe, style='R.TButton',
                               command=lambda: self.move_month(1))
        label = ttk.Label(hframe, width=15, anchor='center')

        hframe.pack(pady=5, anchor=tk.CENTER)
        btn_left.grid(row=0, column=0)
        label.grid(row=0, column=1, padx=12)
        btn_right.grid(row=0, column=2)
        return label

    def move_month(self, offset):
        self.canvas.place_forget()
        month = self.date.month - 1 + offset
        year = self.date.year + month // 12
        month = month % 12 + 1
        self.date = datetime.date(year, month, 1)
        self.build_calendar()

    def create_table(self):
        cols = self.cal.formatweekheader(3).split()
        table = ttk.Treeview(self, show='', selectmode='none',
                             height=7, columns=cols)
        table.bind('<Map>', self.minsize)
        table.pack(expand=1, fill=tk.BOTH)
        table.tag_configure('header', background='grey90')
        table.insert('', tk.END, values=cols, tag='header')
        for _ in range(6):
            table.insert('', tk.END)

        width = max(map(self.font.measure, cols))
        for col in cols:
            table.column(col, width=width, minwidth=width, anchor=tk.E)
        return table

    def minsize(self, e):
        width, height = self.master.geometry().split('x')
        height = height[:height.index('+')]
        self.master.minsize(width, height)

    def create_canvas(self, bg, fg):
        canvas = tk.Canvas(self.table, background=bg,
                           borderwidth=0, highlightthickness=0)
        canvas.text = canvas.create_text(0, 0, fill=fg, anchor=tk.W)
        handler = lambda _: canvas.place_forget()
        canvas.bind('<ButtonPress-1>', handler)
        self.table.bind('<Configure>', handler)
        self.table.bind('<ButtonPress-1>', self.pressed)
        return canvas

    def build_calendar(self):
        year, month = self.date.year, self.date.month
        month_name = self.cal.formatmonthname(year, month, 0)
        month_weeks = self.cal.monthdayscalendar(year, month)

        self.header.config(text=month_name.title())
        items = self.table.get_children()[1:]
        for week, item in zip_longest(month_weeks, items):
            week = week if week else [] 
            fmt_week = ['%02d' % day if day else '' for day in week]
            self.table.item(item, values=fmt_week)

    def pressed(self, event):
        x, y, widget = event.x, event.y, event.widget
        item = widget.identify_row(y)
        column = widget.identify_column(x)
        items = self.table.get_children()[1:]

        if not column or not item in items:
            # clicked te header or outside the columns
            return

        index = int(column[1]) - 1
        values = widget.item(item)['values']
        text = values[index] if len(values) else None
        bbox = widget.bbox(item, column)
        if bbox and text:
            self.selected = '%02d' % text
            self.show_selection(bbox)

    def show_selection(self, bbox):
        canvas, text = self.canvas, self.selected
        x, y, width, height = bbox
        textw = self.font.measure(text)
        canvas.configure(width=width, height=height)
        canvas.coords(canvas.text, width - textw, height / 2 - 1)
        canvas.itemconfigure(canvas.text, text=text)
        canvas.place(x=x, y=y)

    @property
    def selection(self):
        if self.selected:
            year, month = self.date.year, self.date.month
            return datetime.date(year, month, int(self.selected))

def main():
    root = tk.Tk()
    root.title('Tkinter Calendar')
    ttkcal = TtkCalendar(firstweekday=calendar.SUNDAY)
    ttkcal.pack(expand=True, fill=tk.BOTH)
    root.mainloop()

if __name__ == '__main__':
    main()