Certainly not. But then, as you say, that is the righteousness of Christ, not our “own” righteousness. There is no righteousness other than the righteousness of God.
I can’t for the life of me find the NT reference–I was sure it would be in Romans, but I must be too hungry to see it if it’s there. 
I did find the (probable) original OT ref in Isaiah 64, but my concordance (which is usually very good at linking refs up with each other) gives me no NT ref there. Weird.
Anyway. The Isaianic ref is very nifty, especially from an orthodox/evangelical universalistic standpoint; so I thought I would post up the chapter in its entirety. This is from the NASV translation. (Hebrew version treats first verse as ending previous chapter, which in this case I’d have to say is mistaken.)
1.) Oh that Thou wouldst rend the heavens and come down;
that the mountains might quake at Thy presence–
2.) as fire kindles the brushwood; as fire causes water to boil–
to make Thy name known to Thine adversaries!
The nations may tremble at Thy presence!
3.) When Thou didst awesome things which we did not expect,
Thou didst come down, the mountains quaked at Thy presence.
4.) For from of old they [the nations] have not heard nor perceived by ear,
neither has the eye seen a God besides Thee,
Who acts in behalf of the one who waits for Him.
5.) Thou dost meet him who rejoices in doing righteousness,
who remembers Thee in Thy ways.
Behold!–Thou wast angry, for we sinned,
[continuing] in them a long time;
and shall we be saved?!
6.) For all of us have become like one who is unclean,
and all our righteous deeds are like a filthy garment;
and all of us wither like a leaf,
and our iniquities, like the wind, take us away.
7.) And there is no one who calls on Thy name,
who arouses himself to take hold of Thee;
for Thou hast hidden Thy face from us,
and hast delivered [or melted] us
into the power of our iniquities.
[Note, btw, that sinful Israel is put on a par with the Gentiles here.]
8.) But now, O Lord, Thou art our Father!
We are the clay, and Thou our potter,
and all of us are the works of Thy hand!
9.) Do not be angry beyond measure, O Lord,
neither remember iniquity forever:
behold!–look now!–all of us are Thy people!
10.) Thy holy cities have become a wilderness,
Zion has become a wilderness,
Jerusalem a desolation.
11.) Our holy and beautiful house,
where our fathers praised Thee,
has been burned by fire!
And our precious things
have become a ruin.
12.) Wilt Thou restrain Thyself at these things, O Lord?
Wilt Thou keep silent and afflict us beyond measure?!
[The Lord’s answer to this appeal in the final two chapters of Isaiah is well worth reading, too. 
But be sure to follow up the final verses with GosMark 9:49-50, where Jesus directly explains what’s going on in the final verses of Isaiah. 
]
