This isn't an easy read; it's not as clear as it could be. I needed to reread several passages to follow Murray's explanations. He made some points that I haven't read in other material on the topic of remarriage after divorce. I like that he quotes the original Greek, though I can't read it.
Summary
Divorce is biblical in cases of adultery (illicit sex) and when an unbelieving spouse abandons a believer over religious conflict. Remarriage is biblical when both spouses were previously unmarried or biblically divorced. In case of divorce for adultery, Mt 19:9 shows remarriage of innocent party is legitimate. However, there's no biblical evidence that remarriage of guilty party is legitimate.
Murray's position on remarriage after divorce is essentially the position of the Westminster Confession of Faith, except that he believes the allowance for remarriage after divorce in the case of willful desertion applies only when the deserting spouse is an unbeliever.
Notes
Preface
Gen 2:23-24 and Jesus' comments show that marriage bond is originally and ideally indissoluble. Divorce is conceivable only because of rupture of divine-human relations in the Fall.
Cardinal passages: Deut 24:1-4; Matt 5:31-32, 19:3-12; Mk 10:2-12; Lk 16:18; 1 Cor 7:15; Rom 7:1-3.
The Old Testament Provision (Deut 24:1-4)
Deut 24:1-4 doesn't authorize or sanction divorce. It simply says that if a man divorces his wife and she marries another, the former husband can't take her back. However, divorce was practiced, was tolerated and permitted, and was valid under certain circumstances (see Lev 21:7, 14, 22:13; Num 30:9-10; Deut 11:19, 29; Isa 50:1; Jer 3:1; Ezek 44:22).
The bond between the woman and first husband is so sacred that, although divorce may be given and certain freedom granted to divorced persons, and unobliterable relationship appears in the form of unobliterable separation between woman and 1st husband when woman is remarried.
Remarriage itself isn't adultery, and woman isn't called adulteress. Woman and 2nd husband aren't put to death as Pentateuch required for adultery. But remarriage may involve defilement.
What prevents woman from marrying 1st husband isn't that she was subsequently married to another, but that she'd been divorced from 1st husband.
Teaching of Our Lord
Matt 5:31-32
When Jesus in Sermon on Mount says, "it was said … but I say to you," He isn't opposing OT law or abrogating it (Mt 5:17-20). He's contrasting the true intent of OT law with Pharisaical distortions.
Mt 5:31-32 says illicit sex is the only legitimate ground for divorce.
When man divorces for reasons other than illicit sex, woman is illegitimately divorced, and if another man marries her, she and new husband commit adultery.
Only reason 2nd marriage is considered adulterous is because 1st marriage isn't dissolved by illegitimate divorce.
Divorcing husband can't remarry either, because bond isn't broken.
Last part of v 32 refers to woman divorced without legitimate cause, not to all divorced.
This passage doesn't explicitly say whether persons divorced with legitimate cause may remarry.
Jesus abrogated Mosaic death penalty for adultery and legitimized divorce for adultery. Jesus abrogated reasons for divorce given in Deut 24:1-4, tolerated under Mosaic jurisprudence.
Matt 19:3-8
Jesus says that the Mosaic economy permitted or tolerated divorce because of the Israelites' desecration of the divine creation ordinance of marriage, but from the beginning (Creation) there was no such permission.
Matt 19:9
This is only NT passage that combines exceptive clause and remarriage clause.
Matt 19:9 and 5:32 deal with marriage, not betrothal, because they refer to Deut 24:1-4, which deal with marriage.
The man who divorces his wife (except for illicit sex) and marries another is called an adulterer.
Position that remarriage of innocent party is forbidden is held by Roman Catholic Church, Augustine, canon law of Church of England.
Exceptive clause applies to divorce and remarriage, because subject is divorce and remarriage in coordination.
The divorce permitted under the Mosaic economy dissolved the marriage bond. This Mosaic permission is referred to in context of Mt 19:9, 5:31, and Mk 10:2-12, so the divorce described in those places also breaks marriage bond.
What is abrogated is not divorce or its dissolution of marriage bond, but all grounds for divorce except adultery (illicit sex).
If divorce doesn't dissolve marriage bond, then man is putting away his one-flesh wife and failing to perform marital obligations, which is unscriptural.
Mark 10:2-12; Luke 16:18
Jesus' emphasis in Mt 19:3-9 and Mk 10:2-12 is abrogation of permission to divorce for reasons other than adultery, and creating a provision for divorce for reason of adultery.
The fact that Mark and Luke don't mention right of man to divorce wife for adultery doesn't deny that right, nor right for that man to remarry. Since they don't mention right to divorce, it makes sense they also wouldn't mention right to remarry.
Mark and Luke focus on abrogation of Mosaic provisions for divorce, not on situation of adultery. That situation is covered in Matt.
The Teaching of Paul
1 Cor 7:10-15
1 Cor 7:10-11 forbids not only divorce, but even separation of bed and board. But since Mt 5:31 and 19:9 clearly grant right of divorce in case of adultery, 1 Cor 7:10-11 can't intend to ban remarriage after divorce in case of adultery.
Paul's purpose is to show that marriage prevents fornication, and to plead for purity in marriage. In this passage he's not concerned about explaining the rules that apply when marriage is desecrated. Case of adultery is beyond scope of passage.
1 Cor 7:11 propounds no right of separation of dismissal. It can't be used to defend right of separation without dissolution of marriage.
1 Cor 7:11 is saying, "If separation has taken place, let the breach be healed. Failing that, neither party may remarry."
Prior to v 12, Paul is dealing with marriages of 2 Christian spouses. Starting in v 12, he's dealing with mixed marriages of Christian and non-Christian.
V 12-13 say that believing spouse may not leave or put away unbelieving spouse.
V 15 says if the unbelieving spouse departs, the believing spouse is not obligated to pursue, and is freed from all marital debts and duties.
V 15 shows marriage is dissolved. This doesn't conflict with Jesus' teachings because Jesus spoke to question of putting away, but v 15 speaks to willful desertion. Jesus spoke to marriages between believers, but v 15 speaks to mixed marriages. V 10-11 (about 2 Christians) say no remarriage, but v 15 (about mixed marriage) doesn't.
V 15 shows marriage bond is dissolved; the statement is much more decisive and final than v 10-11. Word for "bond" in v 15 is as strong as or stronger than "bound" in 1 Cor 7:27, 39 and Rom 7:2, and in those places it refers to marriage bond.
Position that v 15 shows dissolution of marriage bond isn't the only feasible interpretation.
Westminster Confession of Faith doesn't limit desertion to unbeliever, but it should, to fit 1 Cor 7:15.
Rom 7:1-3
Paul isn't dealing expressly with question of marriage and separation; he's just using marriage as illustration.
This passage gives basic law for marriage: woman is bound to husband as long as he lives. However, if husband desecrates marriage bond by adultery, her relationship to husband is so radically altered without any infringement on her part that she is released from law of husband, and that doesn't violate principle on her part.
For Paul to bring up contingency of adultery would be contrary to principle he's asserting; it would perplex the reader.
Right of dissolution on ground of adultery isn't exception to principle Paul is stating; divorce in case of adultery doesn't violate principle; it introduces new set of conditions under which principle no longer applies to innocent spouse.
The above explanation for Rom 7:2-3 also applies to 1 Cor 7:39.
Practical Cases
There's no biblical warrant for saying a person divorced for adultery (whether innocent or guilty party) commits another act of adultery when remarrying, because the divorce for adultery dissolved marriage bond.
In case of divorce for adultery, Mt 19:9 shows remarriage of innocent party is legitimate. However, there's no biblical evidence that remarriage of guilty party is legitimate. In case of guilty party, although 2nd marriage can't be called illegitimate and adulterous, it also can't be called legitimate because it's not expressly allowed by Scripture.
Murray takes position that when a person remarries after an unbiblical divorce, although 2nd marriage is adulterous and illegitimate, it's a real marriage and binding. However, has "a great deal of respect" for position that in this case, the remarriage is invalid, null and void, and the person should be separated from the 2nd spouse and return to the 1st.