Marriage is a Sacrament.
The Church has nothing to do with a divorce. (That's a civil matter).
The passages in Matthew 19 and Mark 10 are pretty clear about Moses permitting divorce because the peoples' hearts had hardened, while the original law is that "two shall become one flesh" which Jesus pointedly reminds his audience. (You're Doing It Wrong!).
If the Sacrament of Marriage is entered into, then God is by definition involved and the Church by its own rules CANNOT be the instrument of its dissolution. The Church neither supports nor condones divorce in such a case. In that case the Church will not issue a Decree of Nullity.
If the Sacrament of Marriage was not entered into (via a defect of some kind, details in the Code of Canon Law1) then God wasn't in the marriage from the beginning, and the Church can issue a decree of Nullity. It takes a Tribunal to determine this (a non-trivial process) and the two are then free to marry.
In case(4) a civil divorce decree and the Decree of Nullity will match up. In case (3) they won't, and the two are not free to marry (validly, in the church).
The above is consistent with the verses you cite, there is no contradiction.
There is a point cited in Genesis, and in Mark 10 / Matthew 19:5 about how "a man shall leave his father and mother, and the two shall become one flesh." The intent divined from Scripture is that marriage does something profound and permanent.
Matthew 10: 6-9
3 He said to them in reply, "What did Moses command you?"
4 They replied, "Moses permitted him to write a bill of divorce and dismiss her."
5 But Jesus told them, "Because of the hardness of your hearts he wrote you this commandment.
6 But from the beginning of creation, 'God made them male and female.
7 For this reason a man shall leave his father and mother (and be joined to his wife)
8 and the two shall become one flesh.' So they are no longer two but one flesh.
9 Therefore what God has joined together, no human being must separate."
The Catechism expands upon that in addressing the Sacrament of Marriage:
The marriage bond
1639 The consent by which the spouses mutually give and receive one
another is sealed by God himself. From their covenant arises "an
institution, confirmed by the divine law, . . . even in the eyes of
society." The covenant between the spouses is integrated into God's
covenant with man: "Authentic married love is caught up into divine
1640 Thus the marriage bond has been established by God himself in
such a way that a marriage concluded and consummated between baptized
persons can never be dissolved. This bond, which results from the
free human act of the spouses and their consummation of the marriage,
is a reality, henceforth irrevocable, and gives rise to a covenant
guaranteed by God's fidelity. the Church does not have the power to
contravene this disposition of divine wisdom.
The church holds that the two remain married unless as a result of the investigation and ruling of a Tribunal, a defect is uncovered that demonstrates that the sacrament was not in fact entered into/conferred upon the couple.
If the sacrament was conferred, the church will NOT issue a decree of nullity. Divorce is a secular proceeding that won't change the Church's position that the two remain husband and wife.
If the sacrament is found to not have been conferred, due to a defect, then a Decree of Nullity may be issued and the two are free to marry. (The sub text here is that "you weren't validly married if the sacrament was not present").
The core concept is that a marriage is until death do us part. (Or as now said, "as long as we both shall live"). The marriage is viewed as far more than a contract or a ceremony, it is a sacrament, which is generally defined as a visible sign of invisible Grace. In the case of marriage, that Grace is the unification of two people into one. This requires profound selflessness and commitment.
To put it in coarser terms, if you aren't serious about this whole unity thing, you are not entering the marriage honestly. (Dishonest entry into marriage is one of the defects that can be found by a Tribunal considering Nullity decrees). By design, as the Church teaches it, marriage is meant to be a calling of service to each other and to the community. It is one of three Vocation (the two others being priesthood and the vocation of the single life) so the Church encourages those considering marriage to take it seriously. This points to the marriage being a Good Thing(TM) well beyond its goodness for the two getting married. It's good for the whole community. An older teaching presents marriage from a slightly different angle.
"GAUDIUM ET SPES (article 48)"
- The intimate partnership of married life and love has been established by the Creator and qualified by His laws, and is rooted in the jugal covenant of irrevocable personal consent. Hence by that human act whereby spouses mutually bestow and accept each other a relationship arises which by divine will and in the eyes of society too is a lasting one. For the good of the spouses and their
off-springs as well as of society, the existence of the sacred bond no longer depends on human decisions alone. For, God Himself is the author of matrimony, endowed as it is with various benefits and purposes. (St. Augustine, De Bene coniugali PL 40, 375-376 and 394, St. Thomas, Summa Theologica, Suppl. Quaest. 49, art. 3 ad 1, Decretum pro Armenis: Denz.-Schoen. 1327; Pius XI, encyclical letter Casti Connubii: AAS 22 (1930, pp. 547-548; Denz.-Schoen. 3703-3714.) All of these have a very decisive bearing on the continuation of the human race, on the personal development and eternal destiny of the individual members of a family, and on the dignity, stability, peace and prosperity of the family itself and of human society as a whole. By their very nature, the institution of matrimony itself and conjugal love are ordained for the procreation and education of children, and find in them their ultimate crown. Thus a man and a woman, who by their compact of conjugal love "are no longer two, but one flesh" (Matt. 19:ff), render mutual help and service to each other through an intimate union of their persons and of their actions. Through this union they experience the meaning of their oneness and attain to it
with growing perfection day by day. As a mutual gift of two persons, this intimate union and the good of the children impose total fidelity on the spouses and argue for an unbreakable oneness between them. (Pius XI, encyclical letter Casti Connubii: AAS 22 (1930), pp. 546-547; Denz.-Schoen. 3706)
1 Code of Canon Law 1083 to 1094 lists impediments, (I only include some examples):
SPECIFIC DIRIMENT IMPEDIMENTS
Can. 1083 §1. A man before he has completed his sixteenth year of age and a woman before she has completed her fourteenth year of age cannot enter into a valid marriage. §2. The conference of bishops is free to establish a higher age for the licit celebration of marriage.
Can. 1085 §1. A person bound by the bond of a prior marriage, even if it was not consummated, invalidly attempts marriage.
Can. 1087 Those in sacred orders invalidly attempt marriage.
Can. 1088 Those bound by a public perpetual vow of chastity in a religious institute invalidly attempt marriage.
Can. 1089 No marriage can exist between a man and a woman who has been abducted or at least detained with a view of contracting marriage with her unless the woman chooses marriage of her own accord after she has been separated from the captor and established in a safe and free place.
Can. 1090 §1. Anyone who with a view to entering marriage with a certain person has brought about the death of that person’s spouse or of one’s own spouse invalidly attempts this marriage. §2. Those who have brought about the death of a spouse by mutual physical or moral cooperation also invalidly attempt a marriage together.
Can. 1091 §1. In the direct line of consanguinity marriage is invalid between all ancestors and descendants, both legitimate and natural. §2. In the collateral line marriage is invalid up to and including the fourth degree.
§3. The impediment of consanguinity is not multiplied. §4. A marriage is never permitted if doubt exists whether the partners are related by consanguinity in any degree of the direct line or in the second degree of the collateral line.
Can. 1092 Affinity in the direct line in any degree invalidates a marriage.
Can. 1094 Those who are related in the direct line or in the second degree of the collateral line by a legal relationship arising from adoption cannot contract marriage together validly.