Frankly, I always dispute the idea that a Portugal that stays out of Iberian Union would mean there would be no conflict with Netherlands or even England. The Asian spice trade was too profitable for the Dutch not get involved in it instead of only buying it from Portugal. They had resources and the will to avoid the Portuguese middlemen and get more profits by acquiring the products directly from the sources. Why would they simply accept Portuguese monopoly?
A historical study showed that much of Portugal's wealth from the empire of the east prior to 1580 actually flowed northward to the Netherlands (mostly Antwerp). The sale of spices from Lisbon was sent to a Portuguese factory at Antwerp, which acted as a distribution centre for Northern Europe (Germany, the Baltic, Hanseatic League, etc). By 1580, much of this was carried on Flemish ships. In return, loans and insurance were obtained from Flemish financiers along with grains (Lisbon was dependent on imported grains and flour). However, the most important imports were manufactured goods, especially luxury goods such as Flemish tapestries, etc. Finally, the Netherlands acted as a distribution centre for Baltic iron, copper, naval masts, canvas, cordage, armour and gunpowder.
So although Portugal built the fortresses, ships and provided soldiers, sailors and navigators for the empire in Asia, the profits bypassed Lisbon and were going to Flemish, German and Italian bankers. This would eventually benefit the Spanish crown, as it was estimated that Antwerp provided the Spanish crown with seven times as much revenue as all of Spanish America (Urban World History: An Economic and Geographical Perspective by Luc-Normand Tellier).
This commercial relationship led to a cultural exchange between the Netherlands and Portugal. The influence of humanistic thought from the Netherlands began flowing into the Portuguese court beginning in the 1520s. Erasmus for instance had an important influence on Portuguese writers of the day. Flemish art, ceramics, and literature made their way to Portugal as well, for instance the taste blue-white azulejos (tiles) in Portugal originated in the Netherlands during this period, supplanting the earlier designs.
However, with Spanish repression of Protestantism by the 1570s, most of the Protestants from Antwerp went north to Amsterdam. By the time Amsterdam began to eclipse Antwerp, Portugal was under the Spanish Habsburgs, largely excluding it from the trade with the Netherlands, and ending the traditional commercial and cultural exchange between Lisbon and the Netherlands. If Portugal had remained a separate realm, it most likely would have shifted its trading factory to Amsterdam. It would be interesting to see what role Calvinist thought would have had on Portuguese elites.
The Dutch may have still attempted an Asian Empire, along with the English, but it is doubtful their rulers would have encouraged piracy against Portuguese shipping. The Dutch expansion in Asia occurred out of necessity when Philip II cutoff this source of wealth to the Netherlands in an attempt to weaken the United Provinces economically. In OTL, the English and Dutch rulers considered any territory ruled by the Spanish Habsburgs to be free game, so they responded by attacking Portuguese shipping. I imagine Portugal would have been seen as an ally against, and any attempt at expansion in Asia would have been less flagrant.
There were large parts of Asia where Portuguese influence was still tenuous, minimal or non-existent in 1580. If the Dutch still expand in Africa and Asia, they might still build trading factories in places such at the Cape, Java and Sumatra where Portuguese influence was not as strong. However, without a casus belli I just can't imagine them constantly attacking and laying siege to Lisbon and Goa as they did by the 1630s. Also, any expansion in Asia may be directed more towards the Spanish Philippines, Spanish Moluccas, Formosa, etc.
You may also see some interesting new places be colonised by United Provinces and England. In OTL, the English Puritans attempted to establish colonies in St. Augustin in Madagascar in 1645. They may try the Cape (as the Portuguese ignored this region), as the Dutch had not colonised it yet. On, the other hand, Madagascar may become English as a way station to the East. Indochina, Siam and Burma may all become places where the Dutch and English would bypass the Portuguese too.
The Portuguese for there part would have still been unhappy at the loss of the monopoly, but realizing their own weakness, been probably powerless to stop the expansion of these allied or friendly nations. I do see a situation where the Portuguese may still be in control of places like the Malabar Coast, Ceylon, Malacca etc, but the profits flowing largely to Amsterdam and later Bristol, because with peace between the countries and a maintained Anglo-Portuguese alliance, Portugal would have been increasingly dependent on manufactured goods from these countries.