There are 2 categories here: Blessings and Prayers and below that, Calling in the Directions. For longer ones, you'll need to click on the link to get the complete text. If there's a date, it refers to the year we used it in a celebration.  

Blessings and Prayers 

I draw the summer sun into my soul and hold my power high today as Solstice energy courses through my veins and connects me to the ancients, my ancestors, and all who have gone before.
​I am divinely alive.  I feel the warmth of my beloved sun as its rays dance upon my skin on this, the longest day of the year.
Litha Prayer to the Sun from: A Ceremony for Every Occasion: The Pagan Wheel of the Year and Rites of Passage  By Siusaidh Ceanadach:  "The sun is high above us / Shining down upon the land and sea, / Making things grow and bloom. / Great and powerful sun..."  (2017)
Summer Solstice Blessing:  As the sun spirals its longest dance, I bless you. / As nature shows bounty and fertility, I bless you. / May you live with loving intent, / To fulfill your truest destiny.
2000 Ritual Blessing: Yes, you are here in the soft buzzing grass. / Yes, you are listening among the flowering gardens...
So Be It:  
​So be it.  Blessed be.  That which we have seen and said shall come to be.  As we wish it, so shall it be.
Summer Solstice Antiphonal Chant:  What is this night? / It is the night of the Midsummer Solstice...
Summer Solstice Greeting (using sage):  "Today may you embody the spirit of the Empress who rules through love, who’s heart leads the way, who relaxes into her senses and let’s her sexuality flow..." (1 of 3)
Welcome Blessing  
"We greet the sun and you...(name)...as we glory in both of your illuminations, may you feel the power and radiance of your light on this Summer Solstice."
Transformation Prayer:  ​I come to you in prayer, in the mid day light without shadow.  I thank you for lessons...
Do not make any prayer for things to be changed one way or another; just hold these places and beings in your heart and let the sun shine upon them.  Come back to awareness of your own time and place, and give thanks for the longest day.
Become attuned to the midsummer sun.  Absorb the warmth and blessedness, and in return hold in your heart those who lack the blessing of light.
Fairy Blessing of a Place: Through this house give glimmering light / By the dead and drowsy fire: / Every elf and fairy sprite, / Hop as light as bird from brier;...
"Prayer to the Six Powers of the Universe" by Judity Favia, from Mother Earth Spirituality by Ed McGaa, Eagle Man. This is a very powerful and timely prayer to the 6 powers of the universe, West, North, East, South, Mother Earth & Father Sun.  It is a model for a good way to call in the Directions.
Calling in the Directions 
​(Listed is the first stanza so you can get a feel for each.)
So, is there a correct sequence for opening the circle and calling in the Directions?  I've looked at many sites that have varied the order (as have we over the years). As with everything you do in ritual, your intention should guide you.  Anything you use on this site, can be changed to meet your needs.  However, I did find the following guidelines if you choose to use them:  
​"Note that the South is the starting point of the Q’ero Wheel, (South, West, North, East), whereas in the Native American tradition the East is the beginning of the cycle (East, North, West, South)"  (https://www.gaia.com/article/four-directions-ceremony)
We've added the following to our openings and closings:
We begin by acknowledging that our celebration takes place on the ancestral and unceded land of the Mechoopda Indian Tribe of the Maidu People. We pay our respects to elders both past and present, and to their stewarding of the land throughout the generations. We pledge ourselves to respect and honor the earth, our Mother, through our celebration.
(To open the circle:)
By the Earth that is her body,
By the air that is her sweet breath,
By the fire of her bright spirit,
And the water of her living womb,
Our circle is cast, and we are between the worlds. And what happens between the worlds changes the world.
(And to close the circle…)
By the Earth that is her body,
By the air that is her sweet breath,
By the fire of her bright spirit,
And the water of her living womb,
Our circle is open and yet unbroken.
May the peace of the goddess be forever in your heart.
Merry meet and merry part / And merry meet again.
Calling in the Directions 2017:  (Each direction refers to the energy that supports us at different stages of life.)"Spirit Keeper of the East, be with us in the time of birth, of coming to life.  Just as the sun God comes over the horizon, out of the night sky, we too come out from the womb, and into the light of the world..."
Calling in the Directions 2016:  Whispering Wind of the East we stand in stillness, listening, knowing no harm can come to us as we are intent on hearing you.  Spirit of the Wind, carry me, Spirit of the wind carry me home, Spirit of the Wind, carry me home to myself...
Calling in the Circle 2015:  Spirit of the East, Land of the rising Sun, Bringer of Air, the winds that blow across the lands-of new beginning each day and of open horizons. Bring us to the threshold of the mountains we only dare to dream of. Teach us to fly with Eagle, Hawk, and Condor. Oh Great Spirit of the East, we ask that your wisdom and blessing be here with us today...